# Triskeles and Symmetries of Mean Global Sea-Level Pressure

^{*}

## Abstract

**:**

## 1. Introduction

**SLP**).

**SLP**is directly related to climate indices (e.g., [35,57]). Decomposition methods often use “Empirical Orthogonal Functions” (

**EOF**). In this paper, we select an alternate method: the time decomposition of

**SLP**using “Singular Spectrum Analysis” (

**SSA**, e.g., [58], a method that we have used extensively with success in some recent works (e.g., [16,17,18,23,25,27,28].

## 2. The Singular Spectrum Analysis Method

**SSA**was popularized in the geophysical community by the paleoclimatology community in the early 1990s [59,60,61]. These authors developed it with the aim of making up for the limitations of Fourier transform (

**FT**), detecting quasi-periodic components as opposed to perfectly periodic ones in

**FT**, analysis of signals with irregular sampling or mending gaps in the data. Their work was mainly focused on climatic and paleoclimatic series. The method is curiously under-used in our view. We direct the reader interested in knowing all the details of the method to [58], a book with the full, robust mathematical treatment of

**SSA**.

**SSA**can be summarized in four steps. Let us consider a discrete (nonzero) time series (${\mathcal{X}}_{N}$) of length N (N > 2):

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

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

**X**yields a Hankel matrix:

**SSA**, consists in projecting the one-dimensional time series in a multidimensional space of series ${\mathcal{X}}_{N}$ such that vectors ${X}_{i}={({x}_{i},\dots ,{x}_{i+L-1})}^{t}$ belong to ${\mathcal{R}}^{L}$, where $K=N-L+1$. This definition was introduced by Mañé [62] and Takens [63], with the aim of building a space in which strange attractors could be described correctly, often a Banach space. A strange attractor is an object whose dynamical properties may change and evolve into chaos (that is by nature nonlinear). The parameter that controls the embedding is L, the size of the analyzing window, an integer between 2 and $N-1$. The Hankel matrix has a number of symmetry properties: its transpose ${\mathbf{X}}^{t}$, called the trajectory matrix, has the dimension K. Embedding is a compulsory step in the analysis of nonlinear series. It consists formally in the empirical evaluation of all pairs of distances between two offset vectors, delayed (lagged) in order to calculate the correlation dimension of the series. This dimension is rather close to the fractal dimension of strange attractors that could generate that type of series. In this case, it is advised to select for the size of window L very small values (and thus a very large K). On the contrary, for

**SSA**, L must be sufficiently large, so that each vector contains an important part of the information contained in the initial time series. From a mathematical point of view, one must work in the frame of Structural Total Least Squares for a Hankel matrix [64]. This is quite different from the fractal dimension mentioned above. Another benefit of considering a very large L is the possibility of considering sub-vectors ${X}_{i}$ as independent sub-series with distinct dynamics, and thus to be able to identify common characteristics in collections of these sub-series.

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

**SVD**[65] of nonzero 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:

**Remark**

**1.**

**SVD**of the trajectory matrix, as is done in

**SSA**, and multivariate analyses in principal components (

**PCA**) or Karhunen–Loève (

**KL**) decompositions, which are commonly found in time series analysis. SSA differs in the nature of its trajectory matrix, a Hankel matrix with a particular structure: its rows and columns are sub-parts of the signal to be analyzed. They both have a meaning, obviously a temporal physical meaning. Such is not the case for

**PCA**or

**KL**.

**Remark**

**2.**

**SVD**can be replaced by any orthonormal basis (${P}_{i}$). In that case, Equation (6) becomes ${X}_{i}={P}_{i}{Q}_{i}^{t}$ with ${Q}_{i}={X}^{t}{P}_{i}$, for those who want to perform a Fourier analysis with an orthonormal basis made of sine functions. A classical example of an alternate basis is that of the eigenvectors of a self-covariant matrix (Toeplitz SSA).

#### 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, or Hankelization Step)

**Y**be a matrix with the dimensions $L\ast K$ and for each element ${y}_{i,j}$ 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 the kth index of time series y associated with matrix

**Y**gives:

**SSA**of real signals to complex signals. One only has to replace all transposed marks with complex conjugates.

**SSA**(

**iSSA**). Since the 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 obtaining 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 SLP Data

**SLP**data are maintained by the Met Office Hadley Centre https://www.metoffice.gov.uk/hadobs/hadslp2/data/download.html (accessed on 24 October 2020). They are available in map form for global pressure every month since 1850 to the present, with a spatial sampling of $5\xb0\times 5\xb0$. As indicated by the Met Office, the series is not homogeneous in time (two intervals from 1850 to 2004 and 2005 to 2020 in which the means are homogeneous but the variances differ). Allan et al. [66] discuss the locations and number of ground observations they used to build the series. We show the map of mean pressures from 1850 to 2020 in Figure 1.

**SLP**maps, demonstrating the very stable geometry of the global atmospheric structure. Well-known patterns are clearly rendered by the map: in the Southern Hemisphere, the structure is dominated by three large positive features in the southern parts of the main oceans (Pacific, Atlantic and Indian), separated by the southern ends of the three southern continents (South America, South Africa and Australia). This approximate three-fold symmetry actually seems to “leak” into a four-fold symmetry: the positive feature extending from Australia to western South America can be described as exhibiting two weaker features, the one over Australia and east of it being the weakest.

**AAO**), also known as the Southern Annular Mode (

**SAM**), a belt of low pressures surrounding the frozen continent (e.g., [67,68]). A zonal average of Figure 1 is not really consistent with the classical latitudinal description of the Hadley, Ferrel and polar cells, with relatively moderate low pressures over the equatorial belt, high pressures around 30°N and lower values at higher latitudes, particularly over AntarcticaThe overall averaged zonal structure is not symmetrical with respect to the Equator [2].

## 4. The SSA of SLP Data

**iSSA**, e.g., [58]); each time series is decomposed into a sum of a trend and periodic or quasiperiodic components.

**SSA**. It is a sort of mean with respect to time, a nonoscillating function of time; it should not be confused with the spatial mean of maps in time (see, for example, Figure 2).

**SSA**component, represents more than 70% of the total variance (

**sv**) of the original series. The sequence of the next quasi-periodic components is, in decreasing order of periods, ∼130 years (∼0.7 hP, ∼0.06% of the

**sv**; [69]), ∼90 yr (∼21 hP, ∼1.9% of the

**sv**; [70,71]), ∼50 yr (∼0.2 hP, ∼0.02% of the sv), ∼22 yr (∼0.50 hP, ∼0.04% of the

**sv**; Hale cycle, [72]), ∼15 yr ( ∼0.2 hP, ∼0.02% of the

**sv**, upper bound of the Schwabe cycle; [73], ∼4 yr (∼0.3 hP, ∼0.03% of the

**sv**), ∼1.8 yr (∼0.3 hP, ∼0.03% of the

**sv**), then 1 yr ( ∼93 hP, ∼8.3% of the

**sv**), 0.5 yr (∼65 hP, ∼5.8% of the

**sv**), 0.33 yr (∼44 hP, ∼3.9% of the

**sv**) and 0.25 yr (∼21 hP, ∼ 1.9% of the

**sv**) [74].

**SSA**trends of

**SLP**in Hammer–Aitoff and north and south stereographic polar projections. This mean is representative, since the observed (decreasing) overall variation in pressure is only 0.1% in the 170 years of the study. We illustrate the stability of the

**SLP**trend both in space and time with Figure 4, which shows the variation in the trend of

**SLP**since 1850 as a function of latitude. Indeed, the latitudinal structure of mean pressure (

**SSA**trend) is remarkably stable over 170 years (a rather rare feature in geophysics and geodynamics, both internal and external to the solid Earth), with its typical asymmetry between the two hemispheres.

## 5. A Model

**SLP**trend map of Figure 3 (lower left) for the Northern Hemisphere. We invert Equations (14a) and (14b) by simulated annealing [86] for $\omega \ast t$ = 1 (flow stationary and constant in time). The amplitude of the flow perturbation $\epsilon $ is on the order of 0.100 ± 0.002, and the triskeles pattern is centered on 91.3 ± 0.1°W, −77.1 ± 0.2°N. The fit is excellent and compatible with the shearing due to the direction of winds (east to west) at mid latitudes, opposite to the direction of Earth’s rotation from west to east.

**SLP**trend map of Figure 3 (lower right) for the Southern Hemisphere. The amplitude of the flow perturbation $\epsilon $ is reduced to 0.05 ± 0.01, and the symmetry 4 is centered on 88.1 ± 0.2°W, −7.5 ± 0.1°N.

**EOF**or spherical harmonics. On the other hand,

**iSSA**is a scalar time analysis and can extract from the data temporal features. We have shown that atmospheric surface pressures display simple geometric/geographic features that are stable both in space and time. The oscillatory components belong to the harmonic families of the Schwabe (33, 22, 11, 5.5, 3.3 and 1.6 yr) and annual (1, 1/2, 1/3, 1/4 and 1/5 yr) cycles, plus the dominant 90 yr Gleissberg cycle.

## 6. Conclusions

**SLP**since 1850, using the

**iSSA**method of spectral analysis. The method decomposes the original time series into a set of time-dependent components and a trend. The quasi-periodic components are in the decreasing order of periods, cycles of approximately 130, 90, 50, 22, 15, 4, 1.8, 1, 0.5, 0.33 and 0.25 years (see [74] for the harmonic series of 1 year). These are recognized as the Jose, Gleissberg, Hale and Schwabe “long” (decadal and multidecadal) cycles (with some harmonics of the well-known Schwabe “solar” cycle (the Schwabe quasi-cycle is traditionally thought to be 11 yr long but actually spans the 9 to 14 yr range) and the “short” annual cycle and its first three harmonics. These periods are encountered in many solar and terrestrial phenomena and are now attributed to commensurable periods of the Jovian planets [28,29].

**iSSA**are concerned, they have fluctuated by less than 0.1% in 170 years (or 6 × 10

^{−3}hP/yr) studied. The mid latitudes of each hemisphere of the Earth’s surface harbor a set of positive features (on the order of 20 hP), except for one (over Central Asia and Tibet) lying over the main ocean basins (Figure 3, top). This is in full agreement with Schneider et al. [8], who see the oceans as the main engine driving the general circulation of the atmosphere. In stereographic polar projection, the Northern Hemisphere displays a set of three positive features, forming an almost perfect equilateral triangle or triskeles (Figure 3, lower left). The Southern Hemisphere also features a set of three main positive features, but these are arranged as an isosceles triangle, with a possible fourth (but much fainter) feature (Figure 3, lower right). A preliminary analysis suggests that the atmosphere in the two hemispheres could be the site of Taylor–Couette (sheared) differential flow of mode 3 (N hemisphere) or mode 4 (S hemisphere). The mode is imposed by the geometry of the oceanic vs. continental nature of the Earth’s surface. The remarkable regularity and three-order symmetryof the Northern Hemisphere triskeles occur despite the lack of cylindrical symmetry of the northern continents. The stronger intensity and larger size of features in the Southern Hemisphere are due to the presence of the annular flow of cold air over the Southern Ocean and the obstacles due to the southern parts or tips of South America, Southern Africa and Australia.

**SSA**extracts are not artifacts, contrary to what could happen when using spherical harmonics or

**EOF**. A component is present only if there is spatial coherency in the observational data. In the case of sea-level pressure, we find stable (nonoscillatory) components/structures/features (as we have called them). The stability of pressure trends over the 170 years of the analysis implies that the idea of eddies maintaining the cells and forcing their oscillations is no more warranted. Taylor–Couette flow between rotating cylinders provides a good fit to the observations. The geometrical analogy with the cylinders generating the geodynamo may be a promising path to solving the dynamics of the large-scale circulation in the Earth’s atmosphere.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Lorentz, E.N. The Nature and Theory of the General Circulation of the Atmosphere; WMO 218, TP-115; World Meteorological Organization: Geneva, Switzerland, 1967; 161p. [Google Scholar]
- Lindzen, R.S.; Hou, A.V. Hadley circulations for zonally averaged heating centered off the equator. J. Atmos. Sci.
**1988**, 45, 2416–2427. [Google Scholar] [CrossRef] - Hadley, G. VI. Concerning the cause of the general trade-winds. Philos. Trans. R. Soc.
**1735**, 39, 58–62. [Google Scholar] - Ferrel, W. Essay on the winds and ocean currents. Nashv. J. Med. Surg.
**1856**, 11, 287–301. [Google Scholar] - Jeffreys, H. On the dynamics of geostrophic winds. Q. J. R. Meteor. Soc.
**1926**, 52, 85–104. [Google Scholar] [CrossRef] - Starr, V.P. An essay on the general circulation of the earth atmosphere. J. Atmos. Sci.
**1948**, 5, 39–43. [Google Scholar] - Schneider, E.K. Axially symmetric steady-state models of the basic state for instability and climate studies. Part II, Nonlinear calculations. J. Atmos. Sci.
**1977**, 34, 280–296. [Google Scholar] [CrossRef] - Schneider, E.K.; Lindzen, R.S. Axially symmetric steady-state models of the basic state for instability and climate studies. Part I. Linearized calculations. J. Atmos. Sci.
**1977**, 34, 263–279. [Google Scholar] [CrossRef] - 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. Lett.
**2003**, 91, 211101. [Google Scholar] [CrossRef] [PubMed] - Solanki, S.K.; Usoskin, I.G.; Kromer, B.; Schüssler, M.; Beer, J. Unusual activity of the Sun during recent decades compared to the previous 11,000 years. Nature
**2004**, 431, 1084–1087. [Google Scholar] [CrossRef] - Lockwood, M.; Owens, M.J.; Barnard, L.; Davis, C.J.; Steinhilber, F. The persistence of solar activity indicators and the descent of the Sun into Maunder Minimum conditions. Geophys. Res. Lett.
**2011**, 38. [Google Scholar] [CrossRef] - Vieira, L.E.A.; Solanki, S.K.; Krivova, N.A.; Usoskin, I. Evolution of the solar irradiance during the Holocene. Astron. Astrophys.
**2011**, 531, A6. [Google Scholar] [CrossRef] - Abreu, J.A.; Beer, J.; Ferriz-Mas, A.; McCracken, K.G.; Steinhilber, F. Is there a planetary influence on solar activity? Astron. Astrophys.
**2012**, 548, A88. [Google Scholar] [CrossRef] - Kutiev, I.; Tsagouri, I.; Perrone, L.; Pancheva, D.; Mukhtarov, P.; Mikhailov, A.; Lastovicka, J.; Jakowski, N.; Buresova, D.; Blanch, E. et al. Solar activity impact on the Earth’s upper atmosphere. J. Space Weather. Space Clim.
**2013**, 3, A06. [Google Scholar] [CrossRef] [Green Version] - Thuillier, G.; Bolsée, D.; Schmidtke, G.; Foujols, T.; Nikutowski, B.; Shapiro, A.I.; Brunner, R.; Weber, M.; Erhardt, C.; Hersé, M.; et al. The solar irradiance spectrum at solar activity minimum between solar cycles 23 and 24. Sol. Phys.
**2014**, 289, 1931–1958. [Google Scholar] [CrossRef] - Le Mouël, J.L.; Lopes, F.; Courtillot, V. Characteristic time scales of decadal to centennial changes in global surface temperatures over the past 150 years. Earth Space Sci.
**2020**, 7, e2019EA000671. [Google Scholar] [CrossRef] - Le Mouël, J.L.; Lopes, F.; Courtillot, V. Solar turbulence from sunspot records. Mon. Not. R. Astron. Soc.
**2020**, 492, 1416–1420. [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] - Barnes, R.T.H.; Hide, R.; White, A.A.; Wilson, C.A. Atmospheric angular momentum fluctuations, length-of-day changes and polar motion. Proc. R. Soc. A Math. Phys. Sci.
**1983**, 387, 31–73. [Google Scholar] - Rochester, M.G. Causes of fluctuations in the rotation of the Earth. Philos. Trans. R. Soc. Ser. A Math. Phys. Sci.
**1984**, 313, 95–105. [Google Scholar] - Lambeck, K. The Earth’s Variable Rotation: Geophysical Causes and Consequences; Cambridge University Press: Cambridge, UK, 2005. [Google Scholar]
- Schindelegger, M.; Böhm, S.; Böhm, J.; Schuh, H. Atmospheric effects on Earth rotation. In Atmospheric Effects in Space Geodesy; Springer: Berlin/Heidelberg, Germany, 2013; pp. 181–231. [Google Scholar]
- 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] - Barkin, M.Y.; Krylov, S.S.; Perepelkin, V.V. Modeling and analysis of the Earth pole motion with nonstationary perturbations. J. Phys. Conf. Ser.
**2019**, 1301, 012005. [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] - Krylov, S.S.; Perepelkin, V.V.; Filippova, A.S. Long-Period Lunar Perturbations in Earth Pole Oscillatory Process: Theory. In Advances in Theory and Practice of Computational Mechanics: Proceedings of the 21st International Conference on Computational Mechanics and Modern Applied Software Systems; Springer Nature: Singapore, 2020; Volume 173, p. 315. [Google Scholar]
- 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] - 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]
- Lindzen, R.S. Climate dynamics and global change. Annu. Rev. Fluid Mech.
**1994**, 26, 353–378. [Google Scholar] [CrossRef] - Gray, L.J.; Beer, J.; Geller, M.; Haigh, J.D.; Lockwood, M.; Matthes, K.; Cubasch, U.; Fleitmann, D.; Harrison, G.; Hood, L.; et al. Solar influences on climate. Rev. Geophys.
**2010**, 48. [Google Scholar] [CrossRef] - Gray, L.J.; Scaife, A.A.; Mitchell, D.M.; Osprey, S.; Ineson, S.; Hardiman, S.; Butchart, N.; Knight, J.; Sutton, R.; Kodera, K. A lagged response to the 11 year solar cycle in observed winter Atlantic/European weather patterns. J. Geophys. Res. Atmos.
**2013**, 118, 13–405. [Google Scholar] [CrossRef] - Roy, I.; Haigh, J.D. Solar cycle signals in sea level pressure and sea surface temperature. Atmos. Chem. Phys.
**2010**, 10, 3147–3153. [Google Scholar] [CrossRef] - Johnstone, J.A.; Mantua, N.J. Atmospheric controls on northeast Pacific temperature variability and change, 1900–2012. Proc. Natl. Acad. Sci. USA
**2014**, 111, 14360–14365. [Google Scholar] [CrossRef] - 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] - Gruzdev, A.N.; Bezverkhnii, V.A. Manifestation of the 11-year solar cycle in the North Atlantic climate. In Proceedings of the IOP Conference Series: Earth and Environmental Science—Climate Change: Causes, Risks, Consequences, Problems of Adaptation and Management, Moscow, Russia, 26–28 November 2019; Volume 606, p. 012018. [Google Scholar]
- Cionco, R.G.; Kudryavtsev, S.M.; Soon, W.H. Possible Origin of Some Periodicities Detected in Solar-Terrestrial Studies: Earth’s Orbital Movements. Earth Space Sci.
**2021**, 8, e2021EA001805. [Google Scholar] [CrossRef] - Connolly, R.; Soon, W.; Connolly, M.; Baliunas, S.; Berglund, J.; Butler, C.J.; Cionco, R.G.; Elias, A.G.; Fedorov, V.M.; Harde, H.; et al. How much has the Sun influenced Northern Hemisphere temperature trends? An ongoing debate. Res. Astron. Astrophys.
**2021**, 21, 131. [Google Scholar] [CrossRef] - Drews, A.; Huo, W.; Matthes, K.; Kodera, K.; Kruschke, T. The Sun’s Role for Decadal Climate Predictability in the North Atlantic. Atmos. Chem. Phys. Discuss.
**2021**, 22, 7893–7904. [Google Scholar] [CrossRef] - Sonechkin, D.M.; Vakulenko, N.V. Polyphony of Short-Term Climatic Variations. Atmosphere
**2021**, 12, 1145. [Google Scholar] [CrossRef] - Lindzen, R.S. On a calculation of the symmetric circulation and its implications for the role of eddies. Gen. Circ.-Theory Model. Obs.
**1978**, 257–281. [Google Scholar] - Chang, E.K. The influence of Hadley circulation intensity changes on extratropical climate in an idealized model. J. Atmos. Sci.
**1995**, 52, 2006–2024. [Google Scholar] [CrossRef] - Dima, I.M.; Wallace, J.M. On the seasonality of the Hadley cell. J. Atmos. Sci.
**2003**, 60, 1522–1527. [Google Scholar] [CrossRef] - Frierson, D.M.; Lu, J.; Chen, G. Width of the Hadley cell in simple and comprehensive general circulation models. Geophys. Res. Lett.
**2007**, 34. [Google Scholar] [CrossRef] - Hu, Y.; Fu, Q. Observed poleward expansion of the Hadley circulation since 1979. Atmos. Chem. Phys.
**2007**, 7, 5229–5236. [Google Scholar] [CrossRef] - Kharin, V.V.; Zwiers, F.W.; Zhang, X.; Hegerl, G.C. Changes in temperature and precipitation extremes in the IPCC ensemble of global coupled model simulations. J. Clim.
**2017**, 20, 1419–1444. [Google Scholar] [CrossRef] - Lu, J.; Vecchi, G.A.; Reichler, T. Expansion of the Hadley cell under global warming. Geophys. Res. Lett.
**2007**, 34. [Google Scholar] [CrossRef] - Tandon, N.F.; Gerber, E.P.; Sobel, A.H.; Polvani, L.M. Understanding Hadley cell expansion versus contraction: Insights from simplified models and implications for recent observations. J. Clim.
**2013**, 26, 4304–4321. [Google Scholar] [CrossRef] - Shepherd, T.G. Atmospheric circulation as a source of uncertainty in climate change projections. Nat. Geosci.
**2014**, 7, 703–708. [Google Scholar] [CrossRef] - Tao, L.; Hu, Y.; Liu, J. Anthropogenic forcing on the Hadley circulation in CMIP5 simulations. Clim. Dyn.
**2016**, 46, 3337–3350. [Google Scholar] [CrossRef] - Grise, K.M.; Davis, S.M. Hadley cell expansion in CMIP6 models. Atmos. Chem. Phys.
**2020**, 20, 5249–5268. [Google Scholar] [CrossRef] - Schaeffer, M.; Selten, F.M.; Opsteegh, J.D. Shifts of means are not a proxy for changes in extreme winter temperatures in climate projections. Clim. Dyn.
**2005**, 25, 51–63. [Google Scholar] [CrossRef] - Stott, P.A.; Gillett, N.P.; Hegerl, G.C.; Karoly, D.J.; Stone, D.A.; Zhang, X.; Zwiers, F. Detection and attribution of climate change: A regional perspective. Wiley Interdiscip. Rev. Clim. Chang.
**2010**, 1, 192–211. [Google Scholar] [CrossRef] - Rahmstorf, S.; Coumou, D. Increase of extreme events in a warming world. Proc. Natl. Acad. Sci. USA
**2011**, 108, 17905–17909. [Google Scholar] [CrossRef] - Rummukainen, M. Changes in climate and weather extremes in the 21st century. Wiley Interdiscip. Rev. Clim. Chang.
**2012**, 3, 115–129. [Google Scholar] [CrossRef] - Trenberth, K.E.; Fasullo, J.T.; Shepherd, T.G. Attribution of climate extreme events. Nat. Clim. Chang.
**2015**, 5, 725–730. [Google Scholar] [CrossRef] - Vaideanu, P.; Dima, M.; Pirloaga, R.; Ionita, M. Disentangling and quantifying contributions of distinct forcing factors to the observed global sea level pressure field. Clim. Dyn.
**2020**, 54, 1453–1467. [Google Scholar] [CrossRef] - Golyandina, N.; Zhigljavsky, A. Singular Spectrum Analysis for Time Series; Springer: Berlin/Heidelberg, Germany, 2013; Volume 120. [Google Scholar]
- Vautard, R.; Ghil, M. Singular spectrum analysis in nonlinear dynamics, with applications to paleoclimatic time series. Phys. D Nonlinear Phenom.
**1989**, 35, 395–424. [Google Scholar] [CrossRef] - Ghil, M.; Vautard, R. Interdecadal oscillations and the warming trend in global temperature time series. Nature
**1991**, 350, 324–327. [Google Scholar] [CrossRef] - Vautard, R.; Yiou, P.; Ghil, M. Singular-spectrum analysis: A toolkit for short, noisy chaotic signals. Phys. D Nonlinear Phenom.
**1992**, 58, 95–126. [Google Scholar] [CrossRef] - Mañé, R. On the dimension of the compact invariant sets of certain non-linear maps. In Dynamical Systems and Turbulence; Warwick: St. Charles, IL, USA, 1980; Springer: Berlin/Heidelberg, Germany, 1981; pp. 230–242. [Google Scholar]
- Takens, F. Detecting strange attractors in turbulence. In Dynamical Systems and Turbulence; Warwick: St. Charles, IL, USA, 1980; Springer: Berlin/Heidelberg, Germany, 1981; pp. 366–381. [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]
- 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] [Green Version] - Marshall, G.J. Trends in the Southern Annular Mode from observations and reanalyses. J. Clim.
**2003**, 16, 4134–4143. [Google Scholar] [CrossRef] - Gillett, N.P.; Kell, T.D.; Jones, P.D. Regional climate impacts of the Southern Annular Mode. Geophys. Res. Lett.
**2006**, 33. [Google Scholar] [CrossRef] - Jose, P.D. Sun’s motion and sunspots. Astron. J.
**1965**, 70, 193–200. [Google Scholar] [CrossRef] - Gleissberg, W. A long-periodic fluctuation of the sunspot numbers. Observatory
**1939**, 62, 158. [Google Scholar] - 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, 43. [Google Scholar] [CrossRef] - Usoskin, I.G. A history of solar activity over millennia. Living Reviews. Sol. Phys.
**2017**, 14, 1–97. [Google Scholar] - Schwabe, H. Sonnen-Beobachtungen im Jahre 1843. Astron. Nachr.
**1844**, 21, 233–236. [Google Scholar] [CrossRef] - Courtillot, V.; Le Mouël, J.-L.; Lopes, F.; Gibert, D. On the Nature and Origin of Atmospheric Annual and Semi-Annual Oscillations. arXiv
**2022**, arXiv:2206.12133. [Google Scholar] - Chandrasekhar, S. Hydrodynamic and Hydromagnetic Stability; Oxford University Press: Oxford, UK, 1961. [Google Scholar]
- Frisch, U.; Kolmogorov, A.N. Turbulence: The Legacy of AN Kolmogorov; Cambridge University Press: Cambridge, UK, 1995. [Google Scholar]
- Taylor, G.I. VIII. Stability of a viscous liquid contained between two rotating cylinders. Philos. Trans. R. Soc. Ser. A Contain. Pap. Math. Phys. Character
**1923**, 223, 289–343. [Google Scholar] - Schrauf, G. The first instability in spherical Taylor–Couette flow. J. Fluid Mech.
**1986**, 166, 287–303. [Google Scholar] [CrossRef] - Mamun, C.K.; Tuckerman, L.S. Asymmetry and Hopf bifurcation in spherical Couette flow. Phys. Fluids
**1995**, 7, 80–91. [Google Scholar] [CrossRef] - 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] - Mannix, P.M.; Mestel, A.J. Bistability and hysteresis of axisymmetric thermal convection between differentially rotating spheres. J. Fluid Mech.
**2021**, 911, A12. [Google Scholar] [CrossRef] - Forbes, L.K.; Bassom, A.P. Interfacial behaviour in two-fluid Taylor–Couette flow. Q. J. Mech. Appl. Math.
**2018**, 71, 79–97. [Google Scholar] [CrossRef] - Kirkpatrick, S.; Gelatt, C.D.; Vecchi, M.P. Optimization by simulated annealing. Science
**1983**, 220, 671–680. [Google Scholar] [CrossRef] [PubMed] - Gubbins, D.; Bloxham, J. Morphology of the geomagnetic field and implications for the geodynamo. Nature
**1987**, 325, 509–511. [Google Scholar] [CrossRef] - Gubbins, D.; Kelly, P. Persistent patterns in the geomagnetic field over the past 2.5 Myr. Gubbins
**1993**, 365, 829–832. [Google Scholar] [CrossRef]

**Figure 2.**Time average of the spatial pressure gradients from 1850 to 2020 (i.e., winds up to a constant) at sea level (Hammer–Aitoff projection).

**Figure 3.**Mean of trends of pressure at sea level from 1850 to 2020 (

**SLP**), determined using singular spectrum analysis (

**SSA**) (top: Hammer–Aitoff projection; lower left: stereographic projection of the

**NH**; lower right: stereographic projection of the

**SH**).

**Figure 4.**Evolution of the

**SSA**trend of

**SLP**as a function of time (170 yr since 1850) and latitude, illustrating the stability of the zonal pressure structure in time.

**Figure 5.**Solutions of system of Equations (14a) and (14b), for integer mode values q = 1 to 4 (see text).

**Figure 6.**Superimposition of the Taylor–Couette patterns for both North and South Hemispheres. (

**a**) Superimposition of the triskeles’ Taylor–Couette pattern of flow for mode q = 3 and observed mean pattern of sea-level pressure in the North Hemisphere from 1850 to 2020. (

**b**) Superimposition of the Taylor–Couette pattern of flow for mode q = 4 and observed mean pattern of sea-level pressure in the South Hemisphere from 1850 to 2020.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lopes, F.; Courtillot, V.; Le Mouël, J.-L.
Triskeles and Symmetries of Mean Global Sea-Level Pressure. *Atmosphere* **2022**, *13*, 1354.
https://doi.org/10.3390/atmos13091354

**AMA Style**

Lopes F, Courtillot V, Le Mouël J-L.
Triskeles and Symmetries of Mean Global Sea-Level Pressure. *Atmosphere*. 2022; 13(9):1354.
https://doi.org/10.3390/atmos13091354

**Chicago/Turabian Style**

Lopes, Fernando, Vincent Courtillot, and Jean-Louis Le Mouël.
2022. "Triskeles and Symmetries of Mean Global Sea-Level Pressure" *Atmosphere* 13, no. 9: 1354.
https://doi.org/10.3390/atmos13091354