This paper describes a simple method for characterizing global-scale waves in the mesosphere and lower thermosphere (MLT), such as tides and traveling planetary waves, using uniformly gridded two-dimensional longitude–time data. The technique involves two steps. In the first step, the Fourier transform is performed in space (longitude), and then the time series of the space Fourier coefficients are derived. In the second step, the wavelet transform is performed on these time series, and wavelet coefficients are derived. A Fourier–wavelet spectrum can be obtained from these wavelet coefficients, which gives the amplitude and phase of the wave as a function of time and wave period. It can be used to identify wave activity that is localized in time, similar to a wavelet spectrum, but the Fourier–wavelet spectrum can be obtained separately for eastward- and westward-propagating components and for different zonal wavenumbers. The Fourier–wavelet analysis can be easily implemented using existing Fourier and wavelet software. MATLAB and Python scripts are created and made available at

The Earth's atmosphere can support various types of global-scale waves, which zonally extend around a full circle of latitude. The zonal wavenumber is defined as the number of wave cycles that fit within the latitude circle. As the wave propagates eastward or westward, an oscillation is observed at ground stations. The period of the oscillation depends on the zonal phase velocity and zonal wavenumber of the wave,

Examples of global-scale waves in the atmosphere include atmospheric tides

Traveling planetary waves have a period longer than a day and shorter than several weeks. Some are interpreted as normal modes, which are predicted by classical linear wave theory (e.g.,

Traveling planetary waves that are most commonly observed in the MLT region have periods of about 5–7 d

Neither tides nor traveling planetary waves are stationary. Generally, their amplitude varies with season. Besides, tidal amplitude shows marked day-to-day variability in the MLT region (e.g.,

A sudden stratospheric warming is a large-scale meteorological disturbance, which usually takes place in the winter polar stratosphere (e.g.,

Understanding wave activity in the MLT region is important because it has a significant impact on the region above, i.e., the ionosphere and thermosphere (IT; e.g.,

Characterization of global-scale waves requires the identification of the zonal wavenumber and wave period (see Eq.

Assuming that the perturbations of an atmospheric variable

A wavelet analysis is performed in time. The method considered here is the continuous wavelet transform described by

For a given time series

The wavelet transform Eq. (

In a practical application, Eq. (

A wavelet spectrum can be obtained by plotting the amplitude

As described in Sect.

The present study introduces a method to derive global-scale wave spectra, which are similar to those from the CFW analysis. The technique is referred to as Fourier–wavelet analysis without the term “combined” because, in the present approach, the Fourier and wavelet transforms are two independent operations. The Fourier–wavelet technique is easy to implement, using the existing software of Fourier and wavelet transforms, which is readily available in many data analysis software programs such as MATLAB. A Fourier–wavelet spectrum obtained from this analysis gives the amplitude (in units of the input data, unlike a CFW spectrum) and phase of the wave as a function of time and wave period, similar to a wavelet spectrum but separately for eastward- and westward-propagating waves with different zonal wavenumbers.

In the method of

Using Eq. (

In summary, the amplitude

The implementation of the technique is easy, as it requires only standard Fourier and wavelet tools. MATLAB software and Python software (available at

In this section, examples are presented for the application of the Fourier–wavelet analysis to space–time data. The first example uses synthetic data for which the exact wave composition is known. In the other examples, longitude–time data from atmospheric models are analyzed to demonstrate that the technique can be used to identify global-scale waves in the MLT region. For the analysis of atmospheric waves, special attention is paid to sudden stratospheric warming events, where tides and traveling planetary waves in the MLT region often show a large response. The events that are well documented in the literature are selected.

A 2-D data matrix is created that mimics longitude–time data containing global-scale waves. The data, presented in Fig.

The amplitude of wave_A is depicted in Fig.

There was an Antarctic sudden stratospheric warming in September 2019

GAIA model simulation for the period August–October 2019.

Figure

In Fig.

In Fig.

A major Arctic sudden stratospheric warming occurred in January 2009

SD/WACCM-X model simulation for the period January–February 2009.

Observational studies have found large semidiurnal variations in the ionosphere during the January 2009 sudden stratospheric warming

A sudden stratospheric warming that coincides with the spring transition is called a final warning (e.g.,

SD/WACCM-X model simulation for the period January–May 2016.

Figure

As a brief summary, the results presented in Sect.

This study describes a simple method for deriving Fourier–wavelet spectra from 2-D longitude–time data. The method is conceptually similar to that of

Easy-to-use software for computing Fourier–wavelet spectra is created in two user-friendly languages, i.e., MATLAB and Python, and made available at

Future work includes the improvement of the technique for faster computation and broader applications. The technique introduced in this paper relies on the continuous wavelet transform. One criticism against the continuous wavelet transform is that it provides more information than what is actually available under Heisenberg's uncertainty principle

An important limitation of the Fourier–wavelet technique is that it can resolve only global-scale waves. Along with tides and traveling planetary waves, gravity waves are also important in the MLT region (e.g.,

Although this study has focused on waves in the MLT region, the Fourier–wavelet method could be applied to data from other regions of the atmosphere. The technique may also be useful in research areas outside of atmospheric science. The extent of applicability of the technique is still to be explored.

MATLAB and Python software (fourierwavelet v1.1) for computing Fourier–wavelet spectra is available at

The author has declared that there are no competing interests.

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

The author has been supported by the Deutsche Forschungsgemeinschaft (DFG; grant no. YA 574/3-1).

This research has been supported by the Deutsche Forschungsgemeinschaft (grant no. YA 574/3-1).The publication of this article was funded by the Open Access Fund of the Leibniz Association.

This paper was edited by Sylwester Arabas and reviewed by Jun-Ichi Yano and one anonymous referee.