11institutetext: Université Paris Cité, Institut de Physique du globe de
Paris, CNRS UMR 7154, F-75005 Paris, France 22institutetext: LGL-TPE -
Laboratoire de Géologie de Lyon - Terre, Planètes, Environnement, Lyon, France

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

V. Courtillot 11 J-L. Le Mouël 11 F. Lopes 11 D. Gibert 22

This paper proposes a joint analysis of variations of global sea-level
pressure (SLP) and of Earth’s rotation RP, expressed as the coordinates of the
rotation pole ($m_{1}$, $m_{2}$) and length of day (lod). We retain iterative
singular spectrum analysis (iSSA) as the main tool to extract the trend,
periods and quasi periods in the data time series. SLP components are a weak
trend, eleven quasi-periodic or periodic components ($\sim$130, 90, 50, 22,
15, 4, 1.8 yr), an annual cycle and its first three harmonics. These periods
are characteristic of the space-time evolution of the Earth’s rotation axis
(RP) and are present in many characteristic features of solar and terrestrial
physics. The amplitudes of the annual SLP component and its 3 first harmonics
decrease from 93 hPa for the annual to 21 hPa for the third harmonic. In
contrast, the components with pseudo-periods longer than a year range between
0.2 and 0.5 hPa. The trend (21 hPa) could be part of the 90 yr Gleissberg
cycle. We focus mainly on the annual and to a lesser extent semi-annual
components. The annual RP and SLP components have a phase lag of 152 days
(half he Euler period) and they have similar envelopes, offset by about 40
years: polar motion leads atmospheric pressure. Maps of the first three
components of SLP (that together comprise more than 85% of the data variance)
reveal interesting symmetries. The trend is very stable and forms a triskel
structure that can be modeled as Taylor-Couette flow of mode 3. The annual
component is characterized by a large negative anomaly extending over Eurasia
in the NH Summer (and the opposite in the NH Winter) and three large positive
anomalies over Australia and the southern tips of South America and South
Africa in the SH Spring (and the opposite in the SH Autumn) forming a triskel.
The semi-annual component is characterized by three positive anomalies (an
irregular triskel) in the NH Spring and Autumn (and the opposite in the NH
Summer and Winter), and in the SH Spring and Autumn by a strong stable pattern
consisting of three large negative anomalies forming a clear triskel within
the 40∘-60∘ annulus formed by the southern oceans. A large positive anomaly
centered over Antarctica, with its maximum displaced toward Australia, and a
smaller one centered over Southern Africa complement the pattern. Analysis of
SSA components of global sea level pressure shows a rather simple spatial
distribution with the principal forcing factor being changes in parameters of
the Earth’s rotation pole and velocity (measured through length of day). The
flow can probably best be modeled as a set of coaxial cylinders arranged in
groups of three (triskels) or four and controlled by Earth topography and
continent/ocean geography. Flow patterns suggested by maps of the three main
SSA components of SLP (trend, annual and semi-annual) are suggestive of
Taylor-Couette flow. The envelopes of the annual components of SLP and RP are
offset by four decades and there are indications that causality is present in
the sense from changes in Earth rotation axis force pressure variations.

###### Key Words.:

Annual and Semi-annual oscillations, Taylor-Couette, Sea Level Pressure

##  1 Introduction

This paper is an attempt to obtain better constraints on the forcings of the
trend and annual components of both global sea-level pressure and variations
in the Earth’s rotation, and to test the hypothesis that there might be a
causal link between them.

Lopes et al. (2022a) undertook the singular spectral analysis (SSA) of the
evolution of mean sea-level atmospheric pressure (SLP) since 1850. In addition
to dominant trends, a eleven quasi-periodic components were identified, with
(pseudo-) periods of $\sim$130, 90, 50, 22, 15, 4, 1.8, 1, 0.5, 0.33, and 0.25
years, corresponding to the Jose, Gleissberg, Hale and Schwabe cycles, to the
annual cycle and its first three harmonics. These periods are already known to
be characteristic of the space-time evolution of the Earth’s rotation axis:
the rotation pole (RP) also undergoes periodic motions longer than 1 year, at
least up to the Gleissberg $\sim$90 yr cycle (Chandler 1891a, b; Markowitz
1968; Kirov et al. 2002; Lambeck 2005; Zotov et Bizouard 2012; Chao et al.
2014; Zotov et al. 2016; Lopes et al. 2017; Le Mouël et al. 2021; Lopes et al.
2021, 2022a). They are encountered in solar physics (Gleissberg 1939; Jose
1965; Coles et al. 2019; Charvatova et Strestik 1991; Scafetta 2010; Le Mouël
et al. 2017; Usoskin 2017; Scafetta 2020; Courtillot et al. 2021; Scafetta
2021) and terrestrial climate (Wood et al. 1974; Mörth et Schlamminger 1979;
Mörner 1984; Schlesinger et Ramankutty 1994; Lau and Weng 1995; Courtillot et
al. 2007, 2013; Le Mouël et al. 2019a; Scafetta et al. 2020; Connolly et al.
2021).

The rotation velocity is usually expressed as the length of day (lod); it
contains periods of 1 year and shorter, but also (though they are weaker)
longer periods from 11 yr to 18.6 yr (Guinot 1973; Ray et Erofeeva 2014; Le
Mouël et al. 2019b; Dumont et al. 2021; Petrosino et Dumont 1995). Lopes et
al. (2022a) showed that the secular variation of lod since 1846 Stephenson et
Morrison (1984) and RP have been carried by an oscillation whose period fits
the Gleissberg cycle.

RP consists to first order of 3 SSA components that together amount to more
than 85% of the total signal variance Lopes et al. (2017): the Markowitz
(1968) drift, the Chandler (1891a, b) free oscillation (actually a double
component with periods $\sim$433 and $\sim$434 days), and the forced annual
oscillation. Polar drift does not have a universally accepted explanation
(Stoyko 1968; Hulot et al. 1996; Markowitz et Guinot 2022; Deng et al. 2021).
The two other oscillatory components (Chandler and annual) are obtained using
the Liouville-Euler system, that describes the motion of a spherical rotating
solid body (e.g. Lambeck (2005), chapter 3, system 3.2.9).

Polar drift has been discussed in Le Mouël et al. (2021) and Lopes et al.
(2022a) and the free Chandler wobble in Lopes et al. (2021). In the present
paper, we focus on the third major component, the forced annual oscillation.
This oscillation is often assumed to be forced by the Earth’s fluid envelopes,
hence also by climate variations (e.g. Gross et al. (2003), Lambeck (2005)
chapter 7, Bizouard et Seoane (2010); Chen et al. (2019)). The fact that the
observed Chandler period (433-434 days) is not equal to the theoretical value
of the Euler period (306 days) has been interpreted in terms of Earth
elasticity. The annual period is forced by interactions with fluids.
Variations in the Sun-Earth distance, hence of the corresponding gravitational
forces, displace the fluid atmosphere and ocean; exchanges in angular moment
affect RP and lod. The annual oscillation of the rotation pole RP (coordinates
$m_{1}$ and $m_{2}$) would therefore be due to the presence of the fluid
envelopes. If Earth were devoid of fluid envelopes, the annual oscillation
should not exist, which would contradict both the theory and observations of
motion of a Lagrange top (Landau and Lifchitz 1984). Wilson et Haubrich (1976)
recall that Spitaler (1897, 1901) “demonstrated that the annual wobble was
forced, at least in part, by the seasonal migration of air masses on and off
the Asian continent”. Jeffreys (1916) showed that the annual fluctuation in
water storage on the continents was also important. Munk et Mohamed (1961) re-
examined the sources of annual wobble excitation and concluded that the air
mass effect accounted for much but not all of the annual wobble, and water
storage did not explain the remainder. Reservations appear in chapter 7 of
Lambeck (2005), Seasonal Variations, page 146, who writes: “The principal
seasonal oscillation in the wobble is the annual term which has generally been
attributed to a geographical redistribution of mass associated with
meteorological causes. Jeffreys, in 1916, first attempted a detailed
quantitative evaluation of this excitation function by considering the
contributions from atmospheric and oceanic motion, of precipitation, of
vegetation and of polar ice. Jeffreys concluded that these factors explain the
observed annual polar motion, a conclusion still valid today, although the
quantitative comparisons between the observed and computed annual components
of the pole path are still not satisfactory”. In summary, the interpretation
of the forced annual component of polar motion is still hypothetical and fails
to be validated by a numerical model.

In a previous analysis of global sea-level pressure (SLP), we found that
trends since 1950 were very stable in time and space (Lopes et al. 2022b) and
were organized in a dominant 3-fold symmetry about the rotation axis in the
northern hemisphere (NH) and a 3 or 4-fold symmetry in the southern hemisphere
(SH) (Lopes et al. 2022b). These features could be interpreted as resulting
from Taylor-Couette flow. In this paper, we return to the pressure data and
focus on the annual oscillation.

##  2 The Pressure Data and Method of Analysis

The pressure data are maintained by the Met Office Hadley
Centre111https://www.metoffice.gov.uk/hadobs/hadslp2/data/download.html and
can be accessed as maps of global pressure, every month from 1850 up to the
Present. The sampling is 5∘ x 5∘ ; the data we use are labeled HadSLP2r. Allan
et Ansell (2006) have built this series starting with ground observations
whose number and location are shown in their Figure 2. These are subjected to
a quality check, corrected for various local effects then homogenized with the
Empirical Mode Decomposition (EMD) filter.

As in Lopes et al. (2022b), we have used the iterative Singular Spectrum
Analysis algorithm (iSSA) to extract the components, mainly the annual one, in
each grid cell and as a function of time. For the iSSA method, see Golyandina
et Zhigljavsky (2013), for properties of the Hankel and Toeplitz matrices, see
Lemmerling and Van Huffel (2001) and for the Singular Value Decomposition
(SVD) algorithm, see Golub et Reinsch (1971). For summaries of how we use
them, the reader is referred to Lopes et al. (2021).

##  3 The SSA Pressure Components

We obtain the following components:

The trend (from $\sim$ 1009.15 hPa in 1850 to $\sim$ 1008.35 hPa at present)
is the first and largest SSA component. It represents more than 70% of the
total variance of the original series. The sequence of the quasi-periodic
components is, in decreasing order of periods (amplitudes are in hectoPascal,
hPa) :

  * $\bullet$

$\sim$ 130 years ($\sim$0.7 hPa). We recognize the Jose (1965) cycle,

  * $\bullet$

$\sim$ 90 yr ($\sim$21 hPa). We recognize the Gleissberg (1939) cycle (Le
Mouël et al. 2017),

  * $\bullet$

$\sim$ 50 yr ($\sim$0.2 hPa),

  * $\bullet$

$\sim$ 22 yr ($\sim$0.50 hPa). We recognize the Hale cycle (Usoskin et al.
2017),

  * $\bullet$

$\sim$ 15 yr ($\sim$0.2 hPa). We may recognize an upper bound of the Schwabe
(1844), cycle,

  * $\bullet$

$\sim$ 4 yr ($\sim$0.3 hPa),

  * $\bullet$

$\sim$ 1.8 yr ($\sim$0.3 hPa).

Then:

  * $\bullet$

1 yr ( $\sim$93 hPa).

  * $\bullet$

0.5 yr ($\sim$65 hPa).

  * $\bullet$

0.33 yr ($\sim$44 hPa).

  * $\bullet$

0.25 yr ($\sim$21 hPa).

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

Many studies (e.g. Jeffreys (1916); Lambeck (2005); Adhikari et Ivins (2016))
attempt to relate forced (annual) oscillations to polar motion RP ($m_{1}$,
$m_{2}$) or length of day (lod) (data file EOPC01IAU2000, Vondrák et al.
(1995), maintained by IERS222
https://www.iers.org/IERS/EN/DataProducts/EarthOrientationData/eop.html). The
data are shown and discussed in Lopes et al. (2021). The annual SSA components
of $m_{1}$ and $m_{2}$ are displayed in Figure 1. Both undergo some amount of
modulation. That modulation is significant for $m_{1}$, particularly between
1850 and 1930. For $m_{2}$, it is smaller and more regular. A $m_{1}$ vs
$m_{2}$ diagram (see Appendix A) shows the classical Lissajou elliptical shape
of annual polar motion. Note that the date of 1930 is when the Chandler wobble
undergoes a phase jump of $\pi$ (e.g. Gibert (1998); Gibert et Le Mouël
(2008)). The lod data start only in 1962 (not shown, see Lopes et al.
(2022a)). The annual and semi-annual SSA components of lod are modulated
(Figure 2(a) and Figure 2(b)). The annual component grows slightly until 1990,
flattens until 2013 and then starts growing again. The semi-annual component
decreases regularly over the sixty years of available data.

Figure 1: iSSA annual component of rotation pole coordinates $m_{1}$ and
$m_{2}$ from 1850 to the Present.

(a) iSSA annual component of length of day lod from 1850 to the Present

(b) iSSA semi-annual component of length of day lod from 1850 to the Present

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

Figure 3 shows the annual iSSA component since 1850, extracted from the SLP
pressure series. Its modulation looks very much like that of $m_{2}$, with a
phase shift (Figure 1). We have checked that the phase shift is constant over
the 70 years of data. Its mean value is 152.31 $\pm$ 2.68 days. We have
already obtained (exactly) such a phase shift and note that it is (exactly)
half of the Euler period of 306 days: a reason for this remains to be found.
Using the Hilbert transform, we have determined the envelopes of the two
annual components of polar motion and atmospheric pressure (Figure 4(a)). The
polar motion clearly precedes pressure by at least 40 years. On Figure 4(b),
we see that the trend of polar motion is close to the envelope of pressure, in
general preceding it slightly. Given the mathematical properties of the
Liouville-Euler set of equations, the potential direction of causality between
the two phenomena is in the sense of polar motion leading atmospheric
pressure. If correct, this is an important result that confirms Laplace’s
statements in his 1799 Traité de Mécanique Céleste (volume 5, chapter 1, page
347; reproduced in Appendix B).

Thus, after applying the Kepler’s law of conservation of areas, Laplace
concludes that atmospheric motions do not affect pole rotation.

Figure 3: iSSA annual component of sea level pressure SLP from 1850 to the
Present.

(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) Trends of the pole movement (blue curve) and atmospheric pressure SLP
(black curve) extracted by iSSA.

Figure 4: Annual enveloppes and trends extracted from pole movement and SLP

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

We show in Figure 5 polar maps of the mean amplitude of the annual component
(oscillation), for each season since 1850, and in Figure 6 polar maps of the
mean of the semi-annual component.

In Figure 5, in the Spring SH, there is a very regular pattern at the
intersections of the 20∘S to 50∘S annulus and the southern tips of the three
southern continents (Africa, Australia and South America). There is a smaller
dipole with its negative part centered almost perfectly on the South pole, and
its positive part centered on 70∘S, 105∘W. In the Fall, the pattern is the
same but with a reversal in sign. In the Summer, there is a positive band
between 15 and 25°S latitude from Madagascar to New Caledonia (40°-170°E) and
a small negative spot near the South pole (Winter is the same with a sign
reversal). The positive annulus is actually also seen, much weaker, in the
Atlantic and Pacific oceans. The NH is rather different with, in the Spring, a
large negative area over Asia from the Red Sea to Japan and extending in
latitude from 80∘N to 15∘N. This large “anomaly” (using geophysical language)
extends negative arms towards Senegal and over the western USA. Over the year,
the “anomaly” changes sign. Still in the Summer, there are two sizeable
positive anomalies over the northern Atlantic and Pacific Oceans between 40∘N
and 60∘N. In the SH, the large mean annual pressure variations occur over the
Southern continents; in the NH, the Asian continental “anomaly” dominates,
though two significant features occur over the northern parts of the oceans.
Because the Asian “anomaly” and the two (opposite sign) anomalies in the North
Atlantic and Pacific are separated by almost 180° in longitude, their maxima
being always in phase, their contributions to the seasonal excitation function
tend to cancel (Lamb 1972).

The semi-annual component (Figure 6) has stronger patterns of symmetry. The SH
features a very stable 3-fold symmetry (triskel); three large anomalies form
an equilateral triangle centered on 70∘E, 30∘W and 210∘W, between latitudes
40∘S and 60∘S, a large anomaly of opposite sign is centered on the South Pole
(the maximum is actually offset from the SP by 20∘E towards the 120∘E
meridian), and an anomaly with the same sign as that over the pole, but
located over southern Africa. The NH has weaker anomalies forming an irregular
triangle with apices near the North Pacific (40∘N, 165∘E), Baffin Sea (40∘N,
50∘W) and Iran (30∘N, 60∘W).

The bottom two maps in Figure 7 are the Spring maps for the annual and semi-
annual components. We can compare the maps of Figures 5, 6 and 7 for the
annual, semi-annual and trend components with corresponding figures from the
reference treatise of Lamb (1972), reproduced in Appendix C .Recall that taken
together these three components suffice to capture about 85% of the total
signal variance. Figures 4-12 and 4-13 for instance (reproduced in Appendix C
as Figures C1aand Figure C1b) show Lamb’s representation of the average mean
sea level pressure over (respectively) the northern and southern hemispheres
in the 1950s. There is very good agreement for the northern hemisphere (Figure
10(a)): the January map from Lamb and our Winter map show the dominant
“dipole” with the large high pressure (HP) over Asia and the low pressure (LP)
over the northern Pacific. The April and Spring maps do not agree so well. The
July map from Lamb and our Summer map both show a large LP over Asia and two
HPs over the northern Pacific and northern Atlantic. The October map from Lamb
and our Autumn map show a weaker HP over Asia. For the southern hemisphere
(Figure 10(b)), agreement between the two sets of maps is significantly less;
the Winter vs January maps both feature a HP centered on Antarctica and a LP
extending from Madagascar to Australia, and the reverse in Summer/July. Our
Spring (respectively Autumn) maps feature strong HPs (respectively LPs) on the
southern tips of the southern continents. This sharp pattern is not really
seen so well in Lamb’s (1972) maps (Appendix C).

Lamb’s Figure 3-17 (reproduced in Appendix C as Figure C2) represents the
annual mean distribution of atmospheric pressure at sea level for the time
span 1900-40 for the northern hemisphere and 1900-50s for the southern
hemisphere. This can be compared to our maps of the trend (Lopes et al.
(2022b), and Figure 7, top row). The agreement is quite good: the 3 fold
symmetry of the northern hemisphere and the 4 fold symmetry of the southern
hemisphere are even clearer in the Lopes et al. (2022b) maps using the iSSA
method to extract the trend (i.e. the main component) from the data. As
concluded by Lamb (1972) the effects of geography modify the circumpolar
circulation much more in the northern than in the southern hemisphere; “ The
broad zonal character of the long-period mean circulation is as permanent as
the circumpolar arrangement of the heating and cooling zones and the
circumpolar vortex aloft”. It appears to us that such maps are not available
in the published literature; the maps from the southern hemisphere are
particularly interesting, and so is the triskel in Figure 7 (top left).
Specialists may wish to analyze these maps in more detail; they can be
provided on simple request.

(a) During springs

(b) During summers

(c) During autumns

(d) During winters

Figure 5: Polar views of North hemisphere (left column) and South 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 6: Polar views of North hemisphere (left column) and South hemisphere
(right column) showing the mean (1850-2020) of the semi-annual oscillation
(iSSA Component 3) for all (from top to bottom) Springs, Summers, Autumns and
Winters since 1850.

Figure 7: Polar views of North hemisphere (left column) and South hemisphere
(right column) showing the trend (iSSA component 1 from Lopes et al. (2022b),
and maps of iSSAcomponents 2 (annual) and 3 (semi-annual) for all Springs
since 1850.

##  6 Discussion

###  6.1 Symmetries and Forcings

We pointed out in a preliminary paper on symmetries of global sea level
pressure (Lopes et al. 2022b) that the pattern of Figure 7 was remarkably
stable over the 150 years of the data, and that the observed geometry could be
modeled as Taylor-Couette flow of mode 3 (NH) or 4 (SH). The remarkable
regularity and order three symmetry of the NH triskel occurs despite the lack
of cylindrical symmetry of the northern continents. The stronger intensity and
larger size of features in the SH is linked to the presence of the annular
currents. Following Kepler and Laplace (see above) it appears that the
molecules in the atmosphere cannot perturb polar axis motion (“Nous sommes
donc assurés qu’en même temps que les vents analysés diminuent ce mouvement,
les autres mouvements de l’atmosphère qui ont lieu au-delà des tropiques,
l’accélèrent de la même quantité “). The atmosphere behaves as a rotating
cylinder undergoing stationary flow. The top of the cylinder is a free
surface, the bottom is the surface of the solid crust and/or fluid ocean. This
topography interacts strongly with the atmosphere.

###  6.2 Sun-Earth Distance and Phases

The seasonal change in sign agrees with changes in the Sun-Earth distance. The
present values of the ellipticity of the Earth’s orbit and the obliquity of
its rotation axis lead to changes of orbital velocity with Summer in the
southern hemisphere at perihelion and Summer in the northern hemisphere at
aphelion. The torque associated with the Sun changes as velocity changes. As
discussed in Lopes et al. (2021, 2022a), forces acting on the polar axis do so
through a torque ($m_{1}$,$m_{2}$) obeying the Liouville-Euler system of
equations (from Laplace, 1799). The influence of these torques is in
quadrature with the length of day (lod). Changes in orbital velocity of our
planet lead to a variation in the solar torque, which acts against the sense
of rotation of the planet. Thus, annual variations in lod should be in
opposition to changes in the Sun-Earth distance, and they are (Figure 8a,
bottom). In the northern latitudes, both the Earth’s rotation and the winds go
from West to East, the Earth’s rotation slows in Summer and accelerates in
Winter, hence positive pressure anomalies in Winter, negative in Summer. In
other words, the pressure anomalies are due to the variation in rotation
velocity (tracked through lod changes and winds). The phase difference between
the annual components of lod and SLP is constant at 32 $\pm$ 1.2 days (Figure
8b). Relative variations of amplitudes should be similar and indeed they are
(Figure 8b).

(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 $\pm$
1.2 days.

Figure 8: Astronomicals and geophysicals annual oscillations components

##  7 Sketch of a Mechanism

Because more than 70% of the total signal variance is captured by the trend of
the atmospheric pressure, that varies by only 1 per mil since 1850, one can
resort to the theory of turbulent flow in the case of infinitesimal
perturbations (cf. Landau and Lifchitz, 1987). The non stationary $v_{1}(r,t)$
perturbation of the stationary solution $v_{0}(r)$ verifies the Navier-Stokes
equation for an incompressible fluid:

$\dfrac{\partial v_{1}}{\partial
t}+(v_{0}\nabla)v_{1}+(v_{1}\nabla)v_{0}=-\dfrac{p_{1}}{\rho}+\nu.\mathrm{grad}v_{1},\quad\mathrm{div}v_{1}=0$
(1)

The general solution can be written as the sum of particular solutions where
the dependence of $v_{1}$ on time is of the form $e^{-i\omega t}$. There is
still no theoretical basis for a mathematical solution of the stability of
flow about finite dimension bodies plunged in a rotating fluid (eg.
Chandrasekhar (1961); Landau and Lifchitz (1987); Frisch (1995)). Only
experiments show that flow stability depends on the relative value of the
Reynolds number ($R$), each type of flow having its critical Reynolds number
($R_{c}$), beyond which instability sets in (cf. Landau (1944). It is still
difficult to solve turbulent flows other than in the case of concentric
cylinders (e.g. Schrauf (1986); Mamun et Tuckerman (1995); Nakabayashi et
Tsuchida (1995); Hollerbach et al. (2006); Mahloul et al. (2016); Garcia et
al. (2019); Mannix et Mestel (2021)). But fortunately, this is useful in the
case of atmospheric and oceanic flows on the spherical Earth, as is suggested
(among other evidence) by the annular structures shown in Figure 5 to Figure
7.

Rayleigh (1916) tackled the study of stability of a fluid comprised between
two concentric rotating cylinders at high $R$. It was generalized for any
value of $R$ by Taylor (1923) who showed that particular independent solutions
were given by:

$v_{1}(r,\varphi,z)=f(r)*e^{i(n\varphi+kz-\omega t)}$ (2)

$n$ being an integer ($n$=0 for axial symmetry), $k$ the real wavelength of
the instability, and w an acceptable frequency solving the equations in plane
$z$=constant with $v_{1}$=0 on cylinders with radii $r=R_{1}$ and $r=R_{2}$
(see also the paper on triskels by Lopes et al. (2022a). The Reynolds number
is evaluated by $\Omega_{1}R_{1}^{2}/\nu$ (or $\Omega_{2}R_{2}^{2}/\nu$). For
fixed $n$ and $k$, the $\omega$ become a discrete set of $\omega_{n}(k)$
values. In the case of the Earth, $v_{1}$=0 at the solid surface and at the
atmosphere free surface, and the only observed solutions are 24 h and 1 yr. We
therefore expect to find harmonics of these two values, The detailed study for
two concentric cylinders can be found in Chandrasekhar (1961). The $v_{1}$
flow is stationary and consists in toric vortices (or Taylor vortices)
regularly spaced along the cylinder generatrices. If the series of periods
$\omega_{n}(k)$ linked to the annual oscillation is 1, 1/2 1/3, 1/4 yr (see
section 3), then one expects to find experimentally perturbations of order
$n$=3 or 4, as was already the case for trends (Lopes et al. (2022b); Figure 8
top). We find in the present study that indeed such is the case for the annual
(Figure 7 middle) and semi-annual (Figure 7 bottom) oscillations.

##  8 Conclusion

This paper has attempted to obtain better constraints on the sources of the
trend and annual components of both global sea-level pressure (SLP) and
variations in the Earth’s rotation (RP – coordinates of the rotation pole -
and lod – length of day), and to test the hypothesis that there might be a
direct or indirect causal link between them. In a previous study, it had been
shown, using iterative singular spectrum analysis (iSSA), that the mean sea
level atmospheric pressure (SLP) contains, in addition to a weak trend, a
dozen quasi-periodic or periodic components ($\sim$130, 90, 50, 22, 15, 4, 1.8
yr), the annual cycle and its first three harmonics (Lopes et al. 2022a).
These periods were already known to be characteristic of the space-time
evolution of the Earth’s rotation axis (RP) and had been identified in many
characteristic features of solar and terrestrial physics. Polar drift and the
free Chandler wobble having been studied in earlier papers (Le Mouël et al.
2021; Lopes et al. 2022a, 2021), there remained a need of a similar analysis
of the annual and semi-annual components of SLP, RP and lod. Most older and
many recent analyses suggested more or less tentatively that the annual
oscillation in the RP was due to a geographical redistribution of crustal and
atmospheric masses, though it was agreed that quantitative modeling of this
was unsatisfactory (Lambeck 2005).

In the present paper, we have undertaken a more complete analysis of the
annual SSA component of mean sea level global pressure SLP. With the use of
iterative SSA (iSSA), we have extracted the components in each grid cell and
as a function of time since 1850. In section 3, we have listed the components
and their amplitudes (in hPa). We recognize 11 quasi-periodic components plus
the trend. This trend averages 1009 hPa and varies by only 0.7 hPa over the
170 year period with available data. The amplitudes of the annual component
and its 3 first harmonics decrease from 93 hPa for the annual to 21 hPa for
the third harmonic. In contrast, the components with pseudo-periods longer
than a year range between 0.2 and 0.5 hPa. The trend is as large as the annual
(21 hPa) and could be part of the 90 yr Gleissberg cycle.

We have taken the opportunity of this study to further analyze the components
of polar motion RP (coordinates $m_{1}$ and $m_{2}$) and length of day (lod).
Comparing Figure 1 and Figure 3, we see that the annual components of RP and
SLP have a phase difference of 152 $\pm$ 2 days, which is constant over the 70
years of common data; these components are modulated in the same way, growing
in amplitude between $\sim$1870 and $\sim$1960\. We note that the phase
difference happens to be almost exactly half of the Euler period of 306 days,
for reasons we do not know so far. There is actually a phase lag of $\sim$40
years between RP and SLP, as suggested by Figure 4a (showing the envelopes of
the components in Figure 1 and Figure 3) and to a lesser extent by Figure 4b
(showing their first components, the trends). With the Euler-Liouville system
of equations Laplace,1799; Lopes et al. 2021; this paper, Appendix B) in mind,
we hypothesize that if there is a causal link between the two, it is polar
motion that leads atmospheric pressure.

We have mapped the first three iSSA components of global sea level pressure
that account for more than 85% of the total data variance (Figures 5,6 and 7).
The trend had been studied in Lopes et al. (2022b). Its pattern is remarkably
stable over the 150 years of the data, and the observed geometry could be
modeled as Taylor-Couette flow of mode 3 (northern hemisphere - NH or 4
(southern hemisphere - SH). The remarkable regularity and order three symmetry
of the NH triskel occurs despite the lack of cylindrical symmetry of the
northern continents. The stronger intensity and larger size of features in the
SH is linked to the presence of the annular currents. The annual component
(Figure 5) is characterized by a large negative anomaly extending over all of
Eurasia in the NH Summer (and the opposite in the NH Winter) and three large
positive anomalies over Australia and the southern tips of South America and
South Africa in the SH Spring (and the opposite in the SH Autumn). The semi-
annual component (Figure 6) is characterized by three positive anomalies (an
irregular triskel) extending over Iran and the surrounding central Asia, the
North-West Atlantic and the northern Pacific in the NH Spring and Autumn (and
the opposite in the NH Summer and Winter), and in the SH Spring and Autumn by
a strong stable pattern consisting of three large negative anomalies forming a
clear triskel within the 40∘-60∘ annulus formed by the southern oceans
(Central-South Pacific, southern Atlantic and southern Indian Ocean). A large
positive anomaly centered over Antarctica, with its maximum actually displaced
toward Australia, and a smaller one centered over Southern Africa complement
the pattern. The pattern is opposite in the NH Summer and Winter.

In a comparative study of variations in sea level pressure and extent of sea
ice, (Lopes et al. 2022c) checked that the main seasonal forcing of
atmospheric oscillations was the variation of Sun-Earth distance. This is in
full agreement with the changes in sign of the seasonal pressure anomalies
(Figures 5 and 6). In Summer and Winter, annual pressure anomalies are located
essentially in the northern part of the northern hemisphere, whereas in Spring
and Autumn they are located on rough, elevated continental surfaces (central
Asia in the northern hemisphere, the southern tips of South America, south
Africa and all of Australia in the southern hemisphere).

The present values of the ellipticity of the Earth’s orbit and the obliquity
of its rotation axis lead to changes of orbital velocity (Summer in the
southern hemisphere at perihelion and Summer in the northern hemisphere at
aphelion). The torque changes as this velocity changes. As discussed in (Lopes
et al. 2021, 2022a), forces acting on the polar axis do so through the torque
($m_{1}$, $m_{2}$) obeying the Liouville-Euler system of equations. The
influence of this torque is in quadrature with the length of day (lod). Annual
variations in lod are in opposition to changes in the Sun-Earth distance
(Figure 8a, bottom). The pressure anomalies due to the variation in rotation
velocity (tracked through lod changes and winds). The phase difference between
the annual components of lod and SLP is constant at 32 $\pm$ 1.2 days (Figure
8b). Relative variations of amplitudes should be similar and indeed they are
(Figure 8b).

The main features that appear in the maps of Figures 5 to 7 underline the
importance of the geographical patterns of continents vs oceans and the
presence of annular features at latitudes south of 20∘S. Three-fold symmetry
is sometimes centered on or near the poles, mainly in the SH where the annular
ocean ways, the southern tips of three continents and a pole-centered fourth
one force this geometry. We have compared the maps of Figures 5 to 7 (annual,
semi-annual and trend components) to the corresponding figures from Lamb
(1972) (Appendix C). There is in general very good agreement for the northern
hemisphere. For the southern hemisphere, agreement between the two sets of
maps is less. The sharp triskel patterns found in the SSA components are not
seen as well in Lamb’s (1972) maps.Overall, we obtain the same paterns as
Lamb’s (1972) from data that start around 1850 when Lamb is restricted to the
1950’s, this shows the robutsness of these paterns both space and time.

To first order, large scale atmospheric motions can be modeled as rotating
cylinders (Lopes et al. 2022b), stable in time and space, in the shape of 3 or
4 mode triskels (depending on the hemispheres geography). A component of flow
forced by the Sun-Earth distance modifies slightly polar rotation leading to
seasonality of pressure anomalies. Since lod is physically linked to the solid
Earth, geographical regions with strong topography are most affected by these
variations.

It is remarkable that motions are the fully explained in a frame of the theory
of rotating cylinders whitout a need to call for temperature.

## Appendix A

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

(b) Same as in Figure a), with the added dimension of time (1850-2020).

Fig. A: Spatio temporal evolution of the annual component of the rotation
pole.

## Appendix B

Laplace, 1799, Traité de Mécanique Céleste (volume 5, chapter 1, page 347):

“Nous avons fait voir (n∘8), que le moyen mouvement de rotation de la Terre
est uniforme, dans la supposition que cette planète est entièrement solide, et
l’on vient de voir que la fluidité de la mer et de l’atmosphère ne doit point
altérer ce résultat. Les mouvements que la chaleur du Soleil excite dans
l’atmosphère, et d’où naissent les vents alizés semblent devoir diminuer la
rotation de la Terre (…) Mais le principe de conservation des aires, nous
montre que l’effet total de l’atmosphère sur ce mouvement doit être insensible
(…) Nous sommes donc assurés qu’en même temps que les vents analysés diminuent
ce mouvement, les autres mouvements de l’atmosphère qui ont lieu au-delà des
tropiques, l’accélèrent de la même quantité. On peut appliquer le même
raisonnement (…) à tous ce qui peut agiter la Terre dans son intérieur et à sa
surface. Le déplacement de ces parties peut seul altérer ce mouvement; si, par
exemple un corps placé au pole, était transporté à l’équateur ; la somme des
aires devant toujours rester la même, le mouvement de la Terre en serait un
peu diminué; mais pour que cela fut sensible, il faudrait supposer de grands
changement dans la constitution de la Terre”.

## Appendix C

(a) From Lamb (1972) page 156, figure 4-12. Average m.s.l. pressure over the
northern hemisphere, 1950s approx: (a) January, (b) April, (c) July, (d)
October. Prevailing winds blow anticlockwise around the regions of low
pressure, clockwise around high pressure.

(b) From Lamb (1972) page 157, figure 4-13. Average m.s.l. pressure over the
southern hemisphere, 1950s approx: (a) January, (b) April, (c) July, (d)
October. Prevailing winds blow clockwise around the regions of low pressure,
anticlockwise around high pressure.

Fig. C1. Average m.s.l. pressure over northern (top) and the southern
hemisphere (bottom) extracted from Lamb (1972).

(c) Fig. C2: From Lamb (1972) page 113, figure 3-17. Annual mean distribution
of atmospheric pressure in millibars at sea level: (a) northern hemisphere
1900-40 approx, (b) southern hemisphere 1900-40 approx

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