Skip to main content

Isolating spatiotemporally local mixed Rossby-gravity waves using multi-dimensional ensemble empirical mode decomposition


Tropical waves have relatively large amplitudes in and near convective systems, attenuating as they propagate away from the area where they are generated due to the dissipative nature of the atmosphere. Traditionally, nonlocal analysis methods, such as those based on the Fourier transform, are applied to identify tropical waves. However, these methods have the potential to lead to the misidentification of local wavenumbers and spatial locations of local wave activities. To address this problem, we propose a new method for analyzing tropical waves, with particular focus placed on equatorial mixed Rossby-gravity (MRG) waves. The new tropical wave analysis method is based on the multi-dimensional ensemble empirical mode decomposition and a novel spectral representation based on spatiotemporally local wavenumber, frequency, and amplitude of waves. We first apply this new method to synthetic data to demonstrate the advantages of the method in revealing characteristics of MRG waves. We further apply the method to reanalysis data (1) to identify and isolate the spatiotemporally heterogeneous MRG waves event by event, and (2) to quantify the spatial inhomogeneity of these waves in a wavenumber-frequency-energy diagram. In this way, we reveal the climatology of spatiotemporal inhomogeneity of MRG waves and summarize it in wavenumber-frequency domain: The Indian Ocean is dominated by MRG waves in the period range of 8–12 days; the western Pacific Ocean consists of almost equal energy distribution of MRG waves in the period ranges of 3–6 and 8–12 days, respectively; and the eastern tropical Pacific Ocean and the tropical Atlantic Ocean are dominated by MRG waves in the period range of 3–6 days. The zonal wavenumbers mostly fall within the band of 4–15, with Indian Ocean has larger portion of higher wavenumber (smaller wavelength components) MRG waves.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11


Download references


This study has been supported by the National Science Foundation Grants AGS-1139479 and AGS-1723300.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Zhaohua Wu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix: Algorithm for obtaining marginal spectrum of mixed Rossby-Gravity waves

Appendix: Algorithm for obtaining marginal spectrum of mixed Rossby-Gravity waves

The detailed algorithm includes four major steps: (1) use MEEMD to decompose horizontal winds onto spatiotemporally coherent wave fields; (2) use theoretical MRG wave properties to identify MRG wave events; (3) determine spatiotemporally varying wavenumber, frequency, and amplitude of MRG waves using a local sinusoidal fitting method; and (4) synthesize the MRG wave characteristics by projecting MRG wave energy onto a wavenumber-frequency domain to obtain the marginal spectra of different regions. Steps 1 and 4 are introduced in Sect. 2 of the main manuscript. In this appendix, we introduce technical details of steps 2 and 3.

1.1 Identifying MRG wave events

As mentioned in Sect. 2, the identification of an MRG wave event uses four properties of theoretical MRG waves derived by Matsuno (1966): (1) the westward propagating MRG waves have periods ranging from 3 to 12 days if the corresponding wavenumbers are no larger than 20; (2) the horizontal wind fields are equatorially trapped and should resemble the theoretical structure illustrated in Fig. 8a; (3) the westward propagating MRG waves should have a westward phase velocity; and (4) the westward propagating MRG waves should have a eastward group velocity. In this study, all four steps are used for the identified MRG events for the year 2002.

The first criterion leads to us to only screen the MEEMD components 3 and 4 because only these two MEEMD components have periods falling within the theoretically derived period range. The second criterion is the most important for the screening stage; only the wave events hidden in the MEEMD third and fourth components exhibiting alternating meridional wind in the zonal direction with a one-half wavelength larger than 800 km and an amplitude structure close to a Gaussian-distribution in the meridional direction (criterion 2) are considered as potential MRG wave events.

The usage of criterion 3 and 4 is illustrated in Fig. 12, in which the meridional wind at the equator from the fourth component illustrated in Fig. 7a are plotted. The wind at the equator from the fourth component is not very smooth, leading to difficulty in identifying the exact spatial location of a ridge (or trough). To overcome this problem, the fourth component of the meridional wind at the equator (spatial series, lines in Fig. 12a), for any given temporal location, is further decomposed using EEMD. The second component of this new decomposition captures the dominant oscillatory patterns. By tracking the sequential longitudinal locations of a ridge, as illustrated by the bold blue dashed arrow (Fig. 12b), the phase speed is determined. Similarly, by tracking the sequential longitudinal location change of a ridge of the envelope (red curves in Fig. 12b), the group velocity is determined. If a sequential event has a negative phase speed and a positive group velocity, we identify the event as an MRG wave event. Since group velocity of MRG waves is usually small, a minor error may lead to a sign change. In practice, we change the positive group velocity criterion to group velocity greater than − 0.5 ms−1 to tolerate any potential calculation error. The selection of − 0.5 ms−1 is arbitrary, but our sensitivity tests showed that the results in this study is not sensitive to this particular small negative value since large negative group velocity cases rarely occurred in MEEMD components 3 and 4.

Fig. 12
figure 12

The phase speed and group velocity of an MRG wave event. a The sequential meridional winds along Equator from MEEMD component 4 for Oct. 8–12 of 2002, and b their corresponding EEMD component 2’s. The red curves are the envelops of EEMD components 2’s. The blue dashed line indicates the phase speed and the red dashed line the group velocity

1.2 Determining amplitude, local wavenumber, and frequency

According to Matsuno (1966), the meridional wind of MRG wave has the form

$$v = A\exp \left( { - \frac{{y^{2} }}{{L^{2} }}} \right)\cos \left( {kx - \omega t + \varphi } \right),$$

where A is amplitude, y distance from the equator, L the meridional scale, k local zonal wavenumber, and ω local frequency. A completely rational fit to obtain the amplitude requires a three-dimensional surface membrane fit in space of x, y, and t, which presents both technological and computational difficulties. To avoid this obstacle, for any spatiotemporal location at the Equator (x, 0, t), we use three one-dimensional fits: (1) fitting v(y) from a MEEMD component at any temporal and zonal location with a Gaussian curve to obtain Ay; (2) fitting v(t) from a MEEMD component at any temporal and zonal location with a sinusoidal function to obtain At and local frequency ω; and (3) fitting v(x) from a MEEMD component at any temporal and zonal location with a sinusoidal function to obtain Ax and local wavenumber k. The reason we use three different amplitude fitting methods is that for a given temporal at the Equator, there is always a possibility that waves are not at the ridge or the trough phase in either the temporal or equatorial zonal domain. Theoretically, the maxima among (Ay, Ax, At) should be the true amplitude if the fitting domain in each direction covers a sufficiently large domain. The fitting amplitude corresponding to Fig. 6 is presented in Fig. 13.

Fig. 13
figure 13

The MEEMD 4th component of the meridional wind at the equator (the same as Fig. 6) and the fitted amplitude. The values are indicated by the corresponding colorbars with a unit of ms−1. In the right panel, regions with amplitude less than 1.5 ms−1 is left blank

In this study, the meridional grid of data for fitting Ay is from 10°S to 10°N. In sinusoidal fitting for At in the temporal domain, the window size is 7 days for MEEMD component 3 and 20 days for MEEMD component 4. In sinusoidal fitting for Ax in the zonal domain, the window size is 50° in longitude for component 3 and 30° for component 4. All of these fits use the MATLAB fit function. The window sizes are selected as a balance between locality and accuracy. For the purpose of locality, the window size should be as small as possible. For a sinusoidal fitting to have sufficient accuracy, a piece of data containing a positive maximum and negative minimum is a necessary condition; otherwise, the data themselves do not satisfy the minimum form of an oscillatory pattern. Exhaustive tests show the fitted amplitude, frequency, and wavenumber are not sensitive to the minor changes of window sizes, such as meridional window change from 10°S to 10°N to 15°S to 15°N, temporal window in the range of 5–10 days for MEEMD component 3 and 10–25 days for MEEMD component 4, and zonal window size varying by 20%.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Sun, J., Wu, Z. Isolating spatiotemporally local mixed Rossby-gravity waves using multi-dimensional ensemble empirical mode decomposition. Clim Dyn 54, 1383–1405 (2020).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Mixed Rossby-gravity waves
  • Spectra of tropical waves
  • Spatiotemporal inhomogeneity of tropical waves
  • Multi-dimensional ensemble empirical mode decomposition