# On the Nature and Origin of Atmospheric Annual and Semi-Annual Oscillations

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## Abstract

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## 1. Introduction

## 2. The Pressure Data and Method of Analysis

**EMD**) filter.

**iSSA**) to extract the components, mainly the annual one, in each grid cell and as a function of time. For the

**iSSA**method, see [54]; for properties of the Hankel and Toeplitz matrices, see [55]; and for the singular value decomposition (

**SVD**) algorithm, see [56]. We will now summarize the four steps of the SSA method.

#### 2.1. Step 1 (Embedding Step)

**X**with dimension $K\times N$, where $K=N-L+1$ will condition our decomposition. This is the first “tuning knob”. Integrating

**X**yields a Hankel matrix:

#### 2.2. Step 2 (Decomposition in Singular Values **(SVD)**)

**SVD**[56] of non-zero trajectory matrix

**X**(dimensions $L\times K$ ) takes the shape:

**X**such that:

**X**can then be represented as a simple linear sum of elementary matrices ${\mathbf{X}}_{i}$. If all eigenvalues are equal to 1, then decomposition of

**X**into a sum of unitary matrices is:

**SVD**allows one to write

**X**as a sum of d unitary matrices, defined in a univocal way.

**X**be a suite of L lagged parts of ($\mathcal{X}$ and ${X}_{1},\dots ,{X}_{K}$), the linear basis of its eigenvectors. If we let

#### 2.3. Step 3 (Reconstruction)

**SSA**. In order to regroup the unit matrices, one divides the set of indices $i\{1,\dots ,d\}$ into m disjoint subsets of indices $\{{I}_{1},\dots ,{I}_{m}\}$.

#### 2.4. Step 4 (The Diagonal Mean, Aka the Hankelization Step)

**Y**be a matrix with dimension $L*K$ and for each element ${y}_{ij}$ we have $1\u2a7di\u2a7dL$ and $1\u2a7dj\u2a7dK$. Let ${L}^{*}$ be the minimum and ${K}^{*}$ be the maximum. One always has $N=L+K-1$. Finally, let ${y}_{ij}^{*}={y}_{ij}\phantom{\rule{4pt}{0ex}}\mathrm{if}\phantom{\rule{4pt}{0ex}}L<K\phantom{\rule{4pt}{0ex}}\mathrm{and}\phantom{\rule{4pt}{0ex}}{y}_{ij}^{*}={y}_{ji}$ otherwise. The diagonal average applied to kth index of time series y associated with matrix

**Y**gives:

**iSSA**). Because relation (6) is linear, we can iterate the decomposition. We start with a small value of L (we are looking for the longest period) that we increase until getting a quasi-Hankel matrix (step 1 and 2). We then extract the corresponding lowest frequency component that it subtracted from the original signal. We increase again the value of L to find the next component (shortest period). The algorithm stops when no pseudo-cycle can be detected or extracted. In this way, we scan the series from low to high frequencies.

## 3. The SSA Pressure Components

- 1 year ( ∼93 hPa),
- 0.5 years (∼65 hPa),
- 0.33 years (∼44 hPa),
- 0.25 years (∼21 hPa).

## 4. Annual and Semi-Annual SSA Pressure Components and Variations in Polar Motion: Time Analysis

**Figure 2.**Annual and semi-annual components extracted from length of day: (

**a**) iSSA annual component of length of day from 1850 to the present; (

**b**) same as (

**a**) but for the semi-annual component.

## 5. The Annual and Semi-Annual SSA Pressure Components: Spatial Analysis

## 6. Discussion

#### 6.1. Symmetries and Forcings

#### 6.2. Sun–Earth Distance and Phases

## 7. Sketch of a Mechanism

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Spatio temporal evolution of the annual component of the rotation pole: (

**a**) Lissajou pattern of the ${m}_{1}$ vs. ${m}_{2}$ coordinates of the iSSA annual component of the rotation pole; (

**b**) same as in (

**a**), with the added dimension of time (1850–2020).

## References

- Lopes, F.; Courtillot, V.; Le Mouël, J.L.; Gibert, D. Triskeles and Symmetries of Mean Global Sea-Level Pressure. Atmosphere
**2022**, 13, 1354. [Google Scholar] [CrossRef] - Chandler, S.C. On the variation of latitude, I. Astron. J.
**1891**, 11, 59–61. [Google Scholar] [CrossRef] - Chandler, S.C. On the variation of latitude, II. Astron. J.
**1891**, 11, 65–70. [Google Scholar] [CrossRef] - Markowitz, W. Concurrent astronomical observations for studying continental drift, polar motion, and the rotation of the Earth. In Symposium-International Astronomical Union; Cambridge University Press: Cambridge, UK, 1968; Volume 32, pp. 25–32. [Google Scholar]
- Stoyko, A. Mouvement seculaire du pole et la variation des latitudes des stations du SIL. In Symposium-International Astronomical Union; Cambridge University Press: Cambridge, UK, 1968; Volume 32, pp. 52–56. [Google Scholar]
- Kirov, B.; Georgieva, K.; Javaraiah, J. 22-year periodicity in solar rotation, solar wind parameters and earth rotation. In Solar variability: From core to outer frontiers, Proceedings of the 10th European Solar Physics Meeting, Prague, Czech Republic, 9–14 September 2002; Wilson, A., Ed.; ESA SP-506; ESA Publications Division: Noordwijk, The Netherlands, 2002; Volume 1, pp. 149–152. ISBN 92-9092-816-6. [Google Scholar]
- Lambeck, K. The Earth’s Variable Rotation: Geophysical Causes and Consequences; Cambridge University Press: Cambridge, UK, 2005; ISBN 05-2167-330-5. [Google Scholar]
- Zotov, L.; Bizouard, C. On modulations of the Chandler wobble excitation. J. Geodyn.
**2012**, 62, 30–34. [Google Scholar] [CrossRef] - Markowitz, W.; Guinot, B. Continental Drift, Secular Motion of the Pole, and Rotation of the Earth; Springer Science & Business Media: Dordrecht, Holland, 2013; Volume 32. [Google Scholar]
- Chao, B.F.; Chung, W.; Shih, Z.; Hsieh, Y. Earth’s rotation variations: A wavelet analysis. Terra Nova
**2014**, 26, 260–264. [Google Scholar] [CrossRef] - Zotov, L.; Bizouard, C.; Shum, C.K. A possible interrelation between Earth rotation and climatic variability at decadal time-scale. Geod. Geodyn.
**2016**, 7, 216–222. [Google Scholar] [CrossRef] [Green Version] - Lopes, F.; Le Mouël, J.L.; Gibert, D. The mantle rotation pole position. A solar component. Comptes Rendus Geosci.
**2017**, 349, 159–164. [Google Scholar] [CrossRef] - Le Mouël, J.L.; Lopes, F.; Courtillot, V. Sea-Level Change at the Brest (France) Tide Gauge and the Markowitz Component of Earth’s Rotation. J. Coast. Res.
**2021**, 37, 683–690. [Google Scholar] [CrossRef] - Lopes, F.; Le Mouël, J.L.; Courtillot, V.; Gibert, D. On the shoulders of Laplace. Phys. Earth Planet. Inter.
**2021**, 316, 106693. [Google Scholar] [CrossRef] - Lopes, F.; Courtillot, V.; Le Mouël, J.L.; Gibert, D. On two formulations of polar motion and identification of its sources. arXiv
**2022**, arXiv:2204.11611v1. [Google Scholar] [CrossRef] - Gleissberg, W. A long-periodic fluctuation of the sunspot numbers. Observatory
**1939**, 62, 158. [Google Scholar] - Jose, P.D. Sun’s motion and sunspots. Astron. J.
**1965**, 70, 193–200. [Google Scholar] [CrossRef] - Coles, W.A.; Rickett, B.J.; Rumsey, V.H.; Kaufman, J.J.; Turley, D.G.; Ananthakrishnan, S.A.J.W.; Sime, D.G. Solar cycle changes in the polar solar wind. Nature
**1980**, 286, 239–241. [Google Scholar] [CrossRef] - Charvatova, I.; Strestik, J. Long-term variations in duration of solar cycles. Bull. Astron. Inst. Czechoslov.
**1991**, 42, 90–97. [Google Scholar] - Scafetta, N. Empirical evidence for a celestial origin of the climate oscillations and its implications. J. Atmos. Sol.-Terr. Phys.
**2010**, 72, 951–970. [Google Scholar] [CrossRef] [Green Version] - Le Mouël, J.L.; Lopes, F.; Courtillot, V. Identification of Gleissberg cycles and a rising trend in a 315-year-long series of sunspot numbers. Sol. Phys.
**2017**, 292, 1–9. [Google Scholar] [CrossRef] - Usoskin, I.G. A history of solar activity over millennia. Living Rev. Sol. Phys.
**2017**, 14, 1–97. [Google Scholar] [CrossRef] [Green Version] - Scafetta, N. Solar oscillations and the orbital invariant inequalities of the solar system. Sol. Phys.
**2020**, 295, 1–19. [Google Scholar] [CrossRef] - Courtillot, V.; Lopes, F.; Le Mouël, J.L. On the prediction of solar cycles. Sol. Phys.
**2021**, 296, 1–23. [Google Scholar] [CrossRef] - Scafetta, N. Reconstruction of the interannual to millennial scale patterns of the global surface temperature. Atmosphere
**2021**, 12, 147. [Google Scholar] [CrossRef] - Wood, C.A.; Lovett, R.R. Rainfall, drought and the solar cycle. Nature
**1974**, 251, 594–596. [Google Scholar] [CrossRef] - Mörth, H.T.; Schlamminger, L. Planetary motion, sunspots and climate. In Solar-Terrestrial Influences on Weather and Climate; Springer: Dordrecht, The Netherlands, 1979; pp. 193–207. [Google Scholar]
- Mörner, N.A. Planetary, solar, atmospheric, hydrospheric and endogene processes as origin of climatic changes on the Earth. In Climatic Changes on a Yearly to Millennial Basis; Springer: Dordrecht, The Netherlands, 1984; pp. 483–507. [Google Scholar]
- Schlesinger, M.E.; Ramankutty, N. An oscillation in the global climate system of period 65–70 years. Nature
**1994**, 367, 723–726. [Google Scholar] [CrossRef] - Lau, K.M.; Weng, H. Climate signal detection using wavelet transform: How to maketime series sing. Bull. Am. Meteorol. Soc.
**1995**, 76, 2391–2402. [Google Scholar] [CrossRef] - Courtillot, V.; Gallet, Y.; Le Mouël, J.L.; Fluteau, F.; Genevey, A. Are there connections between the Earth’s magnetic field and climate? Earth Planet. Sci. Lett.
**2007**, 253, 328–339. [Google Scholar] [CrossRef] - Courtillot, V.; Le Mouël, J.L.; Kossobokov, V.; Gibert, D.; Lopes, F. Multi-decadal trends of global surface temperature: A broken line with alternating ˜30 years linear segments? Atmos. Clim. Sci.
**2013**, 3, 34080. [Google Scholar] [CrossRef] [Green Version] - Le Mouël, J.L.; Lopes, F.; Courtillot, V. A solar signature in many climate indices. J. Geophys. Res. Atmos.
**2019**, 124, 2600–2619. [Google Scholar] [CrossRef] - Scafetta, N.; Milani, F.; Bianchini, A. A 60-year cycle in the Meteorite fall frequency suggests a possible interplanetary dust forcing of the Earth’s climate driven by planetary oscillations. Geophys. Res. Lett.
**2020**, 47, e2020GL089954. [Google Scholar] [CrossRef] - Connolly, R.; Soon, W.; Connolly, M.; Baliunas, S.; Berglund, J.; Butler, C.J.; Zhang, W. How much has the Sun influenced Northern Hemisphere temperature trends? An ongoing debate. Res. Astron. Astrophys.
**2021**, 21. [Google Scholar] [CrossRef] - Guinot, B. Variation du pôle et de la vitesse de rotation de la Terre. In Traité de Géophysique Interne, Tome I: Sismologie et Pesanteur; Masson & Cie, Éditeurs: Paris, France, 1973. [Google Scholar]
- Ray, R.D.; Erofeeva, S.Y. Long-period tidal variations in the length of day. J. Geophys. Res. Solid Earth
**2014**, 119, 1498–1509. [Google Scholar] [CrossRef] - Le Mouël, J.L.; Lopes, F.; Courtillot, V.; Gibert, D. On forcings of length of day changes: From 9-day to 18.6-year oscillations. Phys. Earth Planet. Inter.
**2019**, 292, 1–11. [Google Scholar] [CrossRef] - Dumont, S.; Silveira, G.; Custódio, S.; Guéhenneux, Y. Response of Fogo volcano (Cape Verde) to lunisolar gravitational forces during the 2014–2015 eruption. Phys. Earth Planet. Inter.
**2021**, 312, 106659. [Google Scholar] [CrossRef] - Petrosino, S.; Dumont, S. Tidal modulation of hydrothermal tremor: Examples from Ischia and Campi Flegrei volcanoes, Italy. Front. Earth Sci.
**2022**, 9, 775269. [Google Scholar] [CrossRef] - Stephenson, F.R.; Morrison, L.V. Long-term changes in the rotation of the Earth: 700 BC to AD 1980. Philos. Trans. R. Soc. A
**1984**, 313, 47–70. [Google Scholar] [CrossRef] - Hulot, G.; Le Huy, M.; Le Mouël, J.L. Influence of core flows on the decade variations of the polar motion. Geophys. Astrophys. Fluid Dyn.
**1996**, 82, 35–67. [Google Scholar] [CrossRef] - Deng, S.; Liu, S.; Mo, X.; Jiang, L.; Bauer-Gottwein, P. Polar drift in the 1990s explained by terrestrial water storage changes. Geophys. Res. Lett.
**2021**, 48, e2020GL092114. [Google Scholar] [CrossRef] - Gross, R.S.; Fukumori, I.; Menemenlis, D. Atmospheric and oceanic excitation of the Earth’s wobbles during 1980–2000. J. Geophys. Res. Solid Earth
**2003**, 108. [Google Scholar] [CrossRef] [Green Version] - Bizouard, C.; Seoane, L. Atmospheric and oceanic forcing of the rapid polar motion. J. Geod.
**2010**, 84, 19–30. [Google Scholar] [CrossRef] - Chen, J.; Wilson, C.R.; Kuang, W.; Chao, B.F. Interannual oscillations in earth rotation. J. Geophys. Res. Solid Earth
**2019**, 124, 13404–13414. [Google Scholar] [CrossRef] - Landau, L.D.; Lifchitz, E.M. Mechanism: Course of Theoretical Physics; Mir Edition: Moscow, Russia, 1984. [Google Scholar]
- Wilson, C.R.; Haubrich, R.A. Meteorological excitation of the Earth’s wobble. Geophys. J. Int.
**1976**, 46, 707–743. [Google Scholar] [CrossRef] [Green Version] - Spitaler, R. Die ursache der breitenschwankungen; Wien, K.-K., Ed.; Hof- und staatsdruckerei: Vienna, Austria, 1897. [Google Scholar]
- Spitaler, R. Die periodischen Luftmassenvershiebungen und ihr Einfluss auf dieLagenanderungen der Erdachse (Breitenschwankungen). Petermanns Mitteilungen Erganxungsband
**1901**, 29, 137. [Google Scholar] - Jeffreys, H. Causes contributory to the Annual Variation of Latitude. (Plate 8). Mon. Not. R. Astron. Soc.
**1916**, 76, 499–525. [Google Scholar] [CrossRef] [Green Version] - Munk, W.; Mohamed, M. Atmospheric excitation of the Earth’s wobble. Geophys. J. Int.
**1961**, 4, 339–358. [Google Scholar] [CrossRef] [Green Version] - Allan, R.; Ansell, T. A new globally complete monthly historical gridded mean sea level pressure dataset (HadSLP2): 1850–2004. J. Clim.
**2006**, 19, 5816–5842. [Google Scholar] [CrossRef] - Golyandina, N.; Zhigljavsky, A. Singular Spectrum Analysis for Time Series; Springer: Berlin, Germany, 2013; ISBN 978-3642349126. [Google Scholar]
- Lemmerling, P.; Van Huffel, S. Analysis of the structured total least squares problem for hankel/toeplitz matrices. Numer. Algorithms
**2001**, 27, 89–114. [Google Scholar] [CrossRef] - Golub, G.H.; Reinsch, C. Singular value decomposition and least squares solutions. In Linear Algebra; Springer: Berlin/Heidelberg, Germany, 1971; pp. 134–151. [Google Scholar]
- Usoskin, I.G.; Solanki, S.K.; Schüssler, M.; Mursula, K.; Alanko, K. Millennium-scale sunspot number reconstruction: Evidence for an unusually active Sun since the 1940s. Phys. Rev. Let.
**2017**, 91, 211101. [Google Scholar] [CrossRef] [Green Version] - Schwabe, H. Sonnenbeobachtungen im Jahre 1843. Von Herrn Hofrath Schwabe in Dessau. Astron. Nachrichten
**1844**, 21, 233. [Google Scholar] - Adhikari, S.; Ivins, E.R. Climate-driven polar motion: 2003–2015. Sci. Adv.
**2016**, 2, e1501693. [Google Scholar] [CrossRef] [Green Version] - Vondrák, J.; Ron, C.; Pesek, I.; Cepek, A. New global solution of Earth orientation parameters from optical astrometry in 1900–1990. Astron. Astrophys.
**1995**, 297, 899. [Google Scholar] - Gibert, D.; Holschneider, M.; Le Mouël, J.L. Wavelet analysis of the Chandler wobble. J. Geophys. Res. Solid Earth
**1998**, 103, 27069–27089. [Google Scholar] [CrossRef] - Gibert, D.; Le Mouël, J.L. Inversion of polar motion data: Chandler wobble, phase jumps, and geomagnetic jerks. J. Geophys. Res. Solid Earth
**2008**, 113. [Google Scholar] [CrossRef] [Green Version] - Laplace, P.S. Traité de Mécanique Céleste; de l’Imprimerie de Crapelet: Paris, France, 1799. [Google Scholar]
- Lamb, H.H. Climate: Present, Past and Future. 1. Fundamentals and Climate Now; Methuen Publishing: London, UK, 1972. [Google Scholar]
- Landau, L.D.; Lifshitz, E.M. Fluid Mechanics: Courses of Theoretical Physics; Pergamon Press: Oxford, UK, 1987; Volume 6. [Google Scholar]
- Chandrasekhar, S. Hydrodynamic and Hydromagnetic Stability; Oxford University Press: Oxford, UK, 1961; ISBN 978-0486640716. [Google Scholar]
- Frisch, U. Turbulence: The Legacy of AN Kolmogorov; Cambridge University Press: Cambridge, UK, 1995; ISBN 0521457130. [Google Scholar]
- Landau, L.D. On the problem of turbulence. Dokl. Akad. Nauk USSR
**1944**, 44, 311. [Google Scholar] - Schrauf, G. The first instability in spherical Taylor-Couette flow. J. Fluid Mech.
**1986**, 166, 287–303. [Google Scholar] [CrossRef] [Green Version] - Mamun, C.K.; Tuckerman, L.S. Asymmetry and Hopf bifurcation in spherical Couette flow. Phys. Fluids
**1995**, 7, 80–91. [Google Scholar] [CrossRef] [Green Version] - Nakabayashi, K.; Tsuchida, Y. Flow-history effect on higher modes in the spherical Couette system. J. Fluid Mech.
**1995**, 295, 43–60. [Google Scholar] [CrossRef] - Hollerbach, R.; Junk, M.; Egbers, C. Non-axisymmetric instabilities in basic state spherical Couette flow. Fluid Dyn Res.
**2006**, 38, 257. [Google Scholar] [CrossRef] - Mahloul, M.; Mahamdia, A.; Kristiawan, M. The spherical Taylor–Couette flow. Eur. J. Mech.-B/Fluids
**2016**, 59, 1–6. [Google Scholar] [CrossRef] - Garcia, F.; Seilmayer, M.; Giesecke, A.; Stefani, F. Modulated rotating waves in the magnetised spherical Couette system. J. Nonlinear Sci.
**2019**, 29, 2735–2759. [Google Scholar] [CrossRef] [Green Version] - Mannix, P.M.; Mestel, A.J. Bistability and hysteresis of axisymmetric thermalconvection between differentially rotating spheres. J. Fluid Mech.
**2021**, 911. [Google Scholar] [CrossRef] - Rayleigh, L. On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side. Lond. Edinb. Dublin Philos. Mag. J. Sci.
**1916**, 32, 529–546. [Google Scholar] [CrossRef] - Taylor, G.I., VIII. Stability of a viscous liquid contained between two rotating cylinders. Philos. Trans. R. Soc. A
**1923**, 223, 289–343. [Google Scholar]

**Figure 1.**The iSSA annual component of rotation pole coordinates ${m}_{1}$ and ${m}_{2}$ from 1850 to the present.

**Figure 4.**Annual envelopes and trends extracted from pole movement and

**SLP**. (

**a**) Envelopes of oscillations of iSSA annual components of polar motion m (blue curve, right scale) and global sea-level pressure

**SLP**(black curve, left scale). (

**b**) Trend of the pole movement (blue curve) and envelope of iSSA component of atmospheric pressure SLP (black curve).

**Figure 5.**Overlay of the triskeles patterns (SSA trend, top) and SSA annual oscillation of SLP (bottom) from Lopes et al. (2022) with maps from Lamb (1972) page 157, Figures 4–13. Left, southern hemisphere; right, southern hemisphere. Lamb’s maps are calculated for a time span of 40 years; our SSA results are for 170 years.

**Figure 6.**Polar stereographic projection of the northern hemisphere (left column) and southern hemisphere (right column) showing the mean (1850–2020) of the annual oscillation (

**iSSA**component 2) for all (from top to bottom) springs, summers, autumns, and winters since 1850: (

**a**) during springs; (

**b**) during summers; (

**c**) during autumns; (

**d**) during winters.

**Figure 7.**Same as Figure 5 but for the semi-annual oscillation (SSA component 3): (

**a**) during springs; (

**b**) during summers; (

**c**) during autumns; (

**d**) during winters.

**Figure 8.**Polar views of northern hemisphere (left column) and southern hemisphere (right column) showing the trend (iSSA component 1 from Lopes et al. [1]), and maps of iSSA components 2 (annual) and 3 (semi-annual) for all springs since 1850.

**Figure 9.**Astronomical and geophysical annual oscillation components: (

**a**) (top) lod data (blue) and their annual component (red); (middle) variation in the Sun–Earth distance (red); (bottom) the two curves above superimposed, showing phase opposition; (

**b**) in green, the annual SSA component 2 of global sea level pressure, and in blue, the annual component of lod. The phase difference is constant at 32 ± 1.2 days.

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**MDPI and ACS Style**

Courtillot, V.; Le Mouël, J.-L.; Lopes, F.; Gibert, D.
On the Nature and Origin of Atmospheric Annual and Semi-Annual Oscillations. *Atmosphere* **2022**, *13*, 1907.
https://doi.org/10.3390/atmos13111907

**AMA Style**

Courtillot V, Le Mouël J-L, Lopes F, Gibert D.
On the Nature and Origin of Atmospheric Annual and Semi-Annual Oscillations. *Atmosphere*. 2022; 13(11):1907.
https://doi.org/10.3390/atmos13111907

**Chicago/Turabian Style**

Courtillot, Vincent, Jean-Louis Le Mouël, Fernando Lopes, and Dominique Gibert.
2022. "On the Nature and Origin of Atmospheric Annual and Semi-Annual Oscillations" *Atmosphere* 13, no. 11: 1907.
https://doi.org/10.3390/atmos13111907