Interannual and intra-seasonal oscillations in the length of day, atmospheric angular momentum and atmospheric surface pressure time series observed at chosen meteorological stations near the equator

The atmospheric surface pressure time series of Madras, Darwin, and Tahiti together with non-tidal length-of-day (LODR) variations and axial component of atmospheric angular momentum (AAM) were analyzed by wavelet transform as well as the combination of the Fourier transform band pass filter with the Hilbert transform (FTBPF+HT) to detect interannual and intra-seasonal oscillations in them. It was found that annual oscillations in the atmospheric surface pressure variations of Darwin and Tahiti stations are in phase and are about 180 o out of phase in the atmospheric surface pressure variations of Madras station. The phase of the annual oscillation in atmospheric surface pressure variations of Madras station is slightly greater (~20 o ) than the phase of the annual oscillation in the LODR time series. The amplitude and phase variations of the annual and semi-annual oscillations computed by the FTBPF+HT combination in LODR and the axial component of AAM are very similar. The mean amplitudes of the semi-annual oscillation in the atmospheric surface pressure variations of Madras and Tahiti are of the order of 0.4 hPa, the phases of these oscillations are stable and the amplitude of the semi-annual oscillation in the atmospheric surface pressure variations of Darwin is negligible due to unstable phase of this oscillation. The atmospheric surface pressure variations of Madras, Darwin, and Tahiti stations show similar amplitude wideband signals with a central period of ~4 years (cutoff periods ranging from about 2.2 to 20 years) related to El Niño phenomenon. The amplitude maxima of this signal corresponding to the strongest El Niño events in 1982-83, 1997-98, and 2014-15 are also present in amplitude variations of this signal in the LODR and AAM χ 3 time series.


Introduction
The Earth's variable rotation including variations of the length of day (LOD), polar motion, precession, and nutation is mostly caused by mass redistributions in different parts of the Earth system and the gravitational torque exerted by the Moon and the Sun. Variations in the Earth's rotation described by polar motion and LODR variations, which are the LOD variations with zonal tide model removed, with respect to an Earth-fixed reference frame are mostly driven by redistribution and motion of masses within and between the atmosphere, the oceans, and land hydrology. The redistribution of mass generates changes in the Earth's inertia tensor, while the mass movement within the fluid layers relative to the solid Earth induces the exchange of angular momentum between these layers and solid Earth. Variations of LODR occur over a wide range of time scales from hours to a few years (e.g. Barnes et al., 1983;Ma et al., 2006a,b) and atmospheric winds and surface pressure changes explain about 90% of the LODR fluctuations (e.g. Gross et al., 2004). Oscillations with periods from about two to a few years in the LODR are excited by large-scale 2-7-year El Niño/Southern Oscillation (ENSO) of the global atmosphere-ocean system (e.g. Chao, 1984;Rosen et al., 1984;Salstein & Rosen, 1986;Dickey et al., 1992;Gambis, 1992). Gipson and Ma (1999) showed that during El Niño event 1982-83 an increase of LODR and the axial component of the atmospheric angular momentum (AAM) was noticed. Dickey et al. (1994) reported the presence of Quasi-Biennial Oscillation (QBO) of the equatorial stratosphere and 4-6 year oscillations related to ENSO in LODR and Chao (1989) showed better agreement QBO+ENSO combination with LODR time series. The annual and semi-annual oscillations excited by the axial component χ 3 of the AAM are the most remarkable components in the LODR changes (e.g. Frostman et al., 1967;Höpfner, 1998).
Long-term observational records of atmospheric surface pressure at some chosen meteorological stations including Madras, Darwin, and Tahiti near the equator, can provide some important information about the relationship between LODR data and surface pressure variations. To investigate relations between the atmosphere surface pressure data at these stations and the LODR time series, the wavelet technique, the Fourier transform band pass filter (FTBPF), and its combination with the Hilbert transform (FTBPF+HT) were applied. The FTBPF+HT algorithms were also applied to investigate relations between LODR and surface pressure variations at Madras, Darwin and Tahiti stations and axial component χ 3 of the AAM time series.

Data
The daily LODR data since 1962.0 till now called eopc04R_IAU2000_daily available at the IERS (International Earth Rotation and Reference Systems Service) website (http://hpiers.obspm.fr/eoppc/series/opa/) was used in the analysis. To extend the LODR time series in the past from 1956.0 to 1962.0 the UT1-TAI data from EOPC01_IAU2000_(1900-now) with a 0.05-year sampling interval were used from which zonal tided model (IERS Conventions 2010) was removed and then UT1R-TAI data were converted to LODR time series.
The monthly atmospheric surface pressure observations monitored from January 1796 to March 2003 at the Madras station (Allan et al., 2002), and monthly atmospheric surface pressure observations at Darwin (since January 1866 till October 2019) and Tahiti meteorological stations and monthly normalized pressure difference (Southern Oscillation Index, SOI) between Tahiti and Darwin since June 1856 till October 2019 (Ropelewski & Jones, 1987) were used in the analysis (http://www.cru.uea.ac.uk/data). The atmospheric surface pressure data at Madras, Darwin, and Tahiti stations are called in this paper as APM, DARW, and TAHI time series, respectively.

Wavelet analysis
Wavelet technique is becoming a common tool for analyzing localized variations of power within a time series. By decomposing a time series into the time-frequency domain, one can determine both the dominant modes of variability and how they vary in time (e.g. Daubechies, 1992;Foufoula-Georgiou & Kumar, 1995Torrence & Compo, 1998, Grinsted et al., 2004. In this paper, we examine LODR, APM, DARW, THAI and SOI time series by continuous wavelet transform (CWT) and to find a possible connection between atmospheric surface pressure time series and LODR, cross wavelet transform (XWT) was applied (Grinsted et al., 2004).
The LODR, APM, DARW, TAHI, and SOI time series are given in the lower panel of Figs. 1-5, respectively. The CWT power spectra of these time series are shown on the upper panel of Figs. 1-5, respectively. The thick black contour in each CWT power spectrum designates the 5% significance level against red noise and the cone of influence (COI). In the case of LODR, there are obvious annual and semi-annual fluctuations with peaks above the 5% significance level (Fig. 1). The most remarkable oscillation in atmospheric surface pressure variations of APM, DARW, and THAI time series is the annual oscillation exceeding the 5% significance level during the whole time of data (Figs. 2-4). The CWT power spectrum of SOI time series computed from the pressure difference between Darwin and Tahiti stations shows significant ~2 to ~10-year oscillations related to ENSO phenomena and it does not show remarkable annual oscillation which is present in DARW and THAI data. The LODR time series from January 1962 to January 2020 (lower) and its CWT power spectrum (upper). The thick black contour designates the 5% significance level against red noise and the cone of influence (COI) where edge effects might distort the picture is shown as a white region.    The CWT power spectra do not show a possible connection between the atmospheric surface pressure at chosen meteorological stations near the equator and LODR time series, so the XWT spectra were computed (Figs. 6-8). The remarkable annual signal above the 5% significance level, shown as a thick contour against red noise, can be noticed between atmospheric surface pressure of the APM, DARW, and THAI data and LODR data during the whole time. The phase differences between the frequency components of the considered time series are shown by the directional arrows in the XWT spectrum. The oscillations are in phase when arrows are pointing right, they are 180 o out of phase when arrows are pointing left and if the second series leads the first one by 90 o the arrows are pointing straight up. It can be noticed that APM is slightly leading LODR during the whole time because the arrows are pointing to the right with a slightly upward direction (Fig. 6). The semi-annual oscillation in the LODR and APM are mostly out of phase because the arrows are generally pointing left (Fig. 6). The annual oscillations in the XWT spectrum of LODR and DARW (Fig. 7) and the XWT spectrum of LODR and TAHI (Fig. 8) time series are mostly ~180 o out of phase. The directions of arrows in the vicinity of the semi-annual oscillation in the XWT spectrum of LODR and DARW are mostly chaotic (Fig. 7) and in the XWT spectrum of LODR and THAI is usually pointing up with a slight left or right direction (Fig. 8). To detect the phase relations between atmospheric surface pressure time series of Darwin and Tahiti with APM data in the whole common data span the XWT spectra between them were computed. The XWT spectra of APM and DARW (Fig. 9) and APM and TAHI (Fig. 10) show that annual oscillations in these pairs are almost ~180 o out of phase because the arrows are pointing left with very slight upward direction. It can be also noticed that these XWT spectra show common signals with periods ranging from ~2 to a dozen years associated with ENSO variability (Figs. 9,10).

Interannual and intra-seasonal oscillations in LODR, AAM and the atmospheric surface pressure time series
First, all atmospheric surface pressure, LODR, and AAM χ 3 time series were interpolated with the sampling interval of 0.02 years (~1 week) using Lagrange interpolating polynomial of degree 2. Such sampling should not influence the filtration results for oscillations with periods greater than a few weeks. The oscillation with central frequency  , computed by the FTBPF is given by the following formula: where ) (t x is a real-valued time series and is the parabolic transmittance function in which λ is half of the window bandwidth and  is the frequency argument. The variable amplitudes and phases of the broadband oscillations can be computed by the FTBPF+HT combination (Kosek et al., 2015(Kosek et al., , 2020: is the complex-valued oscillation with the central frequency  . To filter the annual oscillations from the atmospheric surface pressure APM, DAWR, and TAHI time series, the FTPBF (Eq. 1) was applied with half of the window bandwidth λ=0.004 (cutoff periods are equal to 0.89 and 1.25 years). The annual oscillations in the DARW and THAI time series are in phase and they are both 180 o out of phase with the annual oscillation in the APM time series (Fig. 11) probably due to the opposite direction of Walkers circulation over the Indian and Pacific Oceans. The amplitude and phase variations of the annual oscillation computed by the FTBPF+HT (Eqs. 2 and 3) with half of the filter bandwidth λ=0.004 (cutoff periods are equal to 0.89 and 1.25 years) in the LODR and AAM χ 3 time series are very similar (Fig. 12). It should be emphasized that all calculated amplitude and phase variations show errors at the ends of the time series caused by filtration errors of the FTBPF+HT algorithm. To omit these errors caused by filtration, the filtered series were cut off 1 year at the ends. The phase variations of the annual oscillation in the DARW and THAI time series are very close and the phase variations of the annual oscillation in APM time series are slightly greater than the phase variations of these oscillations in LODR and AAM χ 3 time series (Fig. 12). Zero phase crossings for all the considered time series correspond to the same epoch 1957.0. The amplitudes of the annual oscillation in APM, DARW, and THAI time series are variable and are greater for the APM than for the DARW time series which are greater than for THAI time series (Fig. 12). The variable amplitudes of the semi-annual oscillation computed by the FTBPF+HT with half of the filter bandwidth λ=0.004 (cutoff periods are equal to 0.45 to 0.56 years) in APM and DARW time series are of the same order and are about 2 times greater than in the THAI time series (Fig. 13). The variable phases of the semi-annual oscillations in APM and DARW time series are more stable than the phase of the semi-annual oscillation in the THAI time series due to very small amplitude of this oscillation (Fig.  13).
The mean amplitudes and phases of the annual and semi-annual oscillations together with their standard deviations about the mean are shown in Table  1. The mean amplitude of the annual oscillation in the APM time series in 1957.0 -2002.2 is of the order of 5.1 hPa and the mean amplitudes of the annual oscillation in DARW and THAI time series in 1957.0 -2018.7 are of the order of 3.4 and 1.8 hPa, respectively.
The amplitude and phase variations of the annual and semi-annual oscillations in LODR and AAM χ 3 time series are very similar (Figs. 12,13), however the mean amplitude of the semi-annual oscillation in AAM χ 3 is about 0.05 ms smaller than in LODR time series (Table 1).   The  1982-83, 1997-98, and 2014-15, however other El Niño events in 1972-73 and 2010 could be also noticed (Fig. 14).

Conclusion and discussion
Annual oscillation is remarkable in atmospheric surface pressure observations of Madras (APM), Darwin (DARW), and Tahiti (TAHI) stations as well as LODR, AAM χ 3 time series. The cross wavelet transform (XWT) spectra revealed, a strong link between LODR and APM, DARW, and TAHI data in the annual frequency band. The XWT spectra and the FTBPF+HT combination revealed that annual oscillations in DARW and THAI time series are in phase from 1957 to 2019 and they are ~180 o out of phase with the annual oscillation in APM time series from 1957 to 2003, probably due to opposite direction of Walker circulation over the Indian and Pacific Oceans. The computed by the FTBPF+HT mean phase of the annual oscillation in APM data is about 20 o greater than in the LODR time series and the mean phases of the annual oscillations in DARW and THAI data is ~180 o out of phase with LODR data. It means that the zonal winds and associated surface pressure changes in Madras have a stimulating effect on the annual oscillations in LODR and AAM χ 3 while the zonal winds and the associated surface pressure changes in Tahiti and Darwin weaken this stimulation. Since the annual oscillation amplitude in atmospheric surface pressure variations of Madras (5.1 hPa) is greater than the amplitudes of the annual oscillations of atmospheric surface pressure changes of Tahiti (1.8 hPa) and Darwin (3.4 hPa) then the phase of annual oscillation in AAM χ 3 and LODR is almost similar to the phase of annual oscillation in surface pressure changes of Madras. Thus, the regions of the Indian Ocean where the Madras station is located may have a greater impact on stimulating the annual oscillation in LODR and AAM than the regions of the Pacific Ocean.
The amplitude and phase variations of the annual and semi-annual oscillation in LODR and AAM χ 3 data computed by the FTBPF+HT combination in 1957-2019 are very similar, however, the mean amplitude of the semi-annual oscillation in AAM χ 3 is about 0.05ms smaller than in LODR data. It suggests that seasonal oscillations in LODR are almost entirely excited by the AAM χ 3 time series. The semi-annual oscillations in the atmospheric surface pressure of APM and THAI data were detected by the wavelet transform but under the 5% significance level, however, the FTBPF+HT analysis revealed this oscillation with the amplitude of the order of 0.4 hPa and stable phase. The semi-annual oscillation amplitude in the DARW time series is small and negligible because the phase of this oscillation is very unstable.
The amplitude variations of wideband 4-year (from 2.2 to 20 years) signal in APM, DARW and THAI time series related to ENSO phenomena are similar to the amplitude variations of such oscillations in LODR and AAM χ 3 time series. The maxima of the amplitude of this wideband oscillation correspond to the El Niño events in 1972-73, 1982-83, 1997-98, 2010, and 2014-15 years. The detected ~4-year wideband signal in APM data with similar variability as in DARW and THAI data suggests that atmospheric surface pressure variations at Madras station can be an additional indicator of ENSO variability.