11institutetext:  Université de Paris, 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 annual and semi-annual components of variations in extent of Arctic
and Antarctic sea-ice

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

The time series (1978-2022) of northern hemisphere (NHSI) and southern
hemisphere (SHSI) sea-ice extent are submitted to singular spectral analysis
(SSA). The spectral components are analyzed with Laplace’s formulation of the
Liouville-Euler system of differential equations. Laplace assumes that all
masses on and in Earth are set into motion by astronomical forces. As already
shown in a previous work (see Le Mouël et al., (2021b), figure 03), the trends
observed in the time series are quasi linear, decreasing for NHSI (by 58.300
km2/yr) and increasing for SHSI (by 15.400 km2/yr). The amplitude of annual
variations in the Antarctic is double that in the Arctic, they are in phase
opposition and modulated, with a flat trend for SHSI, a decreasing trend for
NHSI and a sub-decadal modulation of the envelopes. The semi-annual components
are in quadrature. The first three oscillatory components of both NHSI and
SHSI at 1, 1/2 and 1/3 yr account for more than 95% of the signal variance.
The trends are respectively 21 (Antarctic) and 4 times (Arctic) less than the
amplitudes of the annual SSA components. Following Laplace’s views, we
complement previous analyses of variations in pole position (PM for polar
motion, with coordinates $m_{1}$, $m_{2}$) and length of day (lod). Whereas
SSA of lod is dominated by the same first three components as sea-ice, SSA of
PM contains only the 1 yr forced annual oscillation and the Chandler $\sim$1.2
yr components. The 1 yr component of NHSI is in phase with that of lod and in
phase opposition with m1. The reverse holds for the 1 yr component of SHSI. We
note that the semi-annual component appears in lod not in $m_{1}$. The annual
and semi-annual components of NHSI and SHSI are much larger than the trends
observed since 1978, that leads us to test whether a first order geophysical
or astronomical forcing should not be preferred to the mechanisms generally
suggested as a forcing factor of the trends. The lack of modulation of the
largest (SHSI) forced component suggests an alternate mechanism. In Laplace’s
paradigm, the torques exerted by the Moon, Sun and planets play the leading
role as the source of forcing (modulation) of many geophysical phenomena.
These forces (and torques) lead to changes in the inclination of the Earth’s
rotation axis, transferring stresses to the Earth’s solid and fluid envelopes,
setting Earth masses in motion and resulting in thermal dissipation: more than
variations in eccentricity, it is variations in inclination of the rotation
axis that lead to the large annual components of melting and re-freezing of
sea-ice.

###### Key Words.:

sea ice, seasonnal forcing, Taylor-Couette flow

##  1 Introduction

In a series of previous papers, we have applied singular spectrum analysis
(SSA) to long time series of geophysical observables. We have analyzed series
for which several decades to centuries of data were available, allowing one to
explore their periodic or quasi-periodic components in the day to century time
scales. We have successively analyzed polar motion and length of day (Le Mouël
et al. 2019a; Lopes et al. 2021), oceanic climate indices MJO, PDO, ENSO, etc
…(Courtillot et al. 2013; Le Mouël et al. 2019b), global surface temperatures
(Le Mouël et al. 2020a), and in order to contribute to the understanding of
solar-terrestrial relationships sunspot numbers ISSN (Courtillot et al. 2021).
The determination of a comprehensive range of pseudo-periodic components of
ISSN has even allowed us to propose a prediction of the ongoing solar Cycle
25. In the present paper, we extend the same kind of analysis to an important
component of the climate system, namely the variations in surface extent of
sea-ice in the Arctic and Antarctic.

In a previous paper on which the present study is also based, Lopes et al.
(2021) recalled the importance of the treatise of celestial mechanics of
Laplace (1799), and in particular the system of equations, later named after
Euler and Liouville, that express the conservation of kinetic momentum and
govern the declination and inclination of the rotation axis of any planet in
revolution about the Sun (see Appendices A and B). Laplace’s paradigm can be
summarized by two statements: (1) all masses on the surface or internal to a
planet are set in motion by astronomical forces (particularly from the Sun,
Moon and the Jovian planets), and (2) it is sufficient to know the polar
motion or its time derivative (the lod) to have access to the periods of all
moving masses on the surface or inside Earth. This is why the revolution
periods of planets, or their commensurable combinations, and luni-solar tides
have been found in many observed time series of geophysical phenomena, ranging
from hours to millions of years (Milankovic 1920; Mörth & Schlamminger 1978;
Faibridge 1984; Mörner 1984; Laskar et al. 2004; Scafetta 2010; Barnhart &
Eichinger 2011; Manzi et al. 2012; Scafetta 2012; Mörner 2013; Scafetta 2013;
Lopes et al. 2017; Boulila et al. 2018; Le Mouël et al. 2019a, b; Dumont et
al. 2020; Le Mouël et al. 2020a; Lopes et al. 2021; Petrosino & Dumont 2022).

Most recently, Courtillot et al. (2021) have shown that the sunspot number
ISSN, a proxy of solar activity, could be decomposed in a series of
oscillations, all with periods similar to or commensurable with those of
Jovian planets (Mörth & Schlamminger 1978). It has been possible to
reconstruct more than 90% of the variance of the original time series
(Courtillot et al. 2021). Similar results have been obtained for polar motion,
90% of the variance being carried by the ephemerids of the Jovian planets
Lopes et al. (2021). Pioneering work by Milankovic (1920) and more recently by
Laskar et al. (2004) explains how these planets influence over very long times
the obliquity, precession and eccentricity of our planet’s orbit and rotation
axis, and therefore among other consequences climate.

Planets exert gravitational forces (attraction) on all celestial bodies and
because of its mass the Sun is the leading attractor (it exerts a force F of
3.57*1022 kg.m.s-2 on Earth). But because it is close to the center of gravity
of the solar system its contribution to the budget of angular momenta (F*Sun-
Earth distance*Earth’s revolution period = 1.68*1041 kg.m2.s-1) is one to two
orders of magnitude smaller than that of Jovian planets. In a heliocentric
reference frame, Jupiter’s angular momentum is 1.93*1043 kg.m2.s-1, Saturn’s
is 7.82*1042 kg.m2.s-1, Neptune’s 2.50*1042 kg.m2.s-1, and Uranus’s 1.69*1042
kg.m2.s-1 .

We present the data for the North and South hemisphere sea-ice extents (NHSI
and SHSI respectively) in section 2 and their spectral components using both
Fourier analysis and SSA in section 3. We provide complementary results on
polar motion and lod in section 4. We compare the annual components of polar
motion and lod with those of sea-ice extent in section 5. A hypothesis is
discussed in section 6 and we close with a conclusion in section 7.

##  2 Available data for sea-ice extent in both hemispheres

Very precise daily series of sea-ice extent in the Arctic (NHSI) and Antarctic
(SHSI) are available, starting in 1978
111ftp://sidads.colorado.edu/pub/DATASETS/NOAA/G02135/south/daily/data. The
data set (Cavalieri & Parkinson 2012; Fetterer et al. 2017) from which the
NHSI and SHSI sea-ice areas are calculated consists of sea ice concentration
maps derived from the radiances obtained from microwave radiometers on a suite
of satellites. This started with the Nimbus 7 Scanning Multichannel Microwave
Radiometer (SMMR), which operated from 1978 through 1987, then the Defense
Meteorological Satellite Program (DMSP) series of F8, F11, F13 and F15 Special
Sensor Microwave Imagers (SSMI), and the F17 Special Sensor Microwave Imager
Sounder (SSMIS). The F8 operated from 1987 through 1991, the F11 from 1991
through 1995, the F13 from 1995 through 2007, and the F17 from 2008 through
2010 (Cavalieri et al. (1997, 1999, 2011), Cavalieri & Parkinson (2012)).
After 2010 and up to 2018, we refer to Parkinson (2019), quoting Cavalieri et
al. (1999, 2011).

The data set is generated using the Advanced Microwave Scanning Radiometer -
Earth Observing System (AMSR-E) Bootstrap Algorithm, with daily varying tie-
points. Daily (every other day prior to July 1987) and monthly data are
available for both the north and south polar regions. Data are gridded on the
SSM/I polar stereographic grid (25 x 25 km) and provided in two-byte integer
format. The sampling from 10/26/1978 to 12/02/1987 is not as continuous as
afterwards; there is only one datum every other day from 10/26/1978 to
08/20/1987, one per day from 08/20/1987 to 12/02/1987, then a gap from
12/02/1987 and 01/13/1988, back to one point per day from 01/13/1988 to
02/19/2022, a two day gap from 02/19/2022 to 02/22/2022, and finally full
sampling (one per day) until 03/21/2022. We have worked with the data set as
described above, starting in 10/26/1978 and ending in 03/21/2022
(https://nsidc.org/data/G02135/versions/3). The data are shown in Figure
(1(a)) (NHSI in blue, SHSI in red).

A number of features are immediately apparent from this figure: annual
variations are dominant, their amplitude in the Antarctic is double that in
the Arctic, they are in phase opposition and modulated, with a flat trend for
SHSI, a decreasing trend for NHSI and a sub-decadal modulation of the
envelopes of both. Figure (2(a)) shows the Fourier spectra of the two series
with successive enlargements: a fundamental annual period is accompanied by 7
harmonics from 1/2 to 1/8 yr.

(a) Raw southern hemisphere (SHSI, red) and northern hemisphere (NHSI, blue)
sea-ice data (https://nsidc.org/data/G02135/versions/3).

(a) Fourier spectrum of the SHSI (red) and NHSI (blue) series of Figure
(1(a)). Figure 2: Temporal evolution of SHSI(red) and NHSI (blue)

##  3 SSA analysis of the sea-ice extent data NHSI and SHSI

We next apply SSA (Singular spectral analysis; Golyandina et al. (2018)) as in
our previous papers (e.g. Lopes et al. (2021)) in order to determine the
waveforms (amplitude and phase) of the periodic or quasi-periodic components
of the sea-ice extent series. As seen in Figure 2, they are dominated by the
modulation of a quasi-monochromatic cycle.

For NHSI, the first three components allow one to recover 98.7% of the total
signal variance. They occur at periods 1.000 $\pm$ 0.014 yr (95.6% of the
variance), 0.499 $\pm$ 0.003 yr (2.69%), and 0.333 $\pm$ 0.001 yr (0.43%). For
SHSI the first three components allow one to recover 97.9% of the total signal
variance. They occur at periods 0.999 $\pm$ 0.014 yr (94.4%), 0.500 $\pm$
0.004 yr (3.25%), and 0.333 $\pm$ 0.001 yr (0.25%). Remaining components have
a negligible contribution. In the following, we focus on the annual and semi-
annual components that are displayed in Figure (3(a)) and(4(a))respectively.

The annual Antarctic sea-ice component is only slightly modulated on a multi-
decadal time scale, whereas the annual Arctic component is modulated, with a
significant growth. The two are in phase opposition (180 $\pm$ 1 day) as
expected. For the semi-annual components, the phase difference is 34 $\pm$ 2
days, which is close to quadrature (90/2 days).

The trends are quasi linear (not shown in the figures), decreasing for NHSI
(by 58.300 km2/yr) and increasing for SHSI (by 15.400 km2/yr). The total
surface change since 1978 is a gain of 0.57*106 km2 for the Antarctic and a
loss of 2.15*106 km2 for the Arctic. These values are in good agreement with
Parkinson (2019). Note that they are respectively 21 (Antarctic) and 4 times
(Arctic) less than the amplitudes of the annual SSA components.

(a) SSA annual component of SHSI (red) and NHSI (blue) from 1978 to 2022 (top)
and their phase difference (bottom; in days).

(a) SSA semi-annual component of SHSI (red) and NHSI (blue) from 1978 to 2022
(top), an enlargement from 2010 to 2022 (middle) and their phase difference
(bottom, in days). Figure 4: Annual and semi-annual components extracted from
SHSI(red) and NHSI (blue)

##  4 Complementary results for polar motion and length of day

The orientation of the Earth’s rotation axis is on many time scales a key
variable that modulates the insolation of any location on the Earth’s surface.
It is therefore interesting to compare the spectral components of sea-ice
extent with the variations in polar motion (PM) and length of day (lod). These
data are available on the site of the International Celestial Reference System
(IERS)222https://www.iers.org/IERS/EN/DataProducts/EarthOrientationData/eop.html,
where we have selected file EOP14C04. Several authors have studied the
spectral content of lod. In (Le Mouël et al. 2019a) we explored with SSA the
range from 9.13 days to 18.6 years. Longer periods at 18.6 yr, 11 yr, and some
harmonics, and shorter periods at 27 days and harmonics are linked to the Sun
and Moon, and are discussed elsewhere (Le Mouël et al. 2019a). For this paper,
we focus on the range below 2 years (Figure 5; Fourier spectrum). Distinct
peaks are seen at 1 and 1/2 yr, but also at 1/3 and 1/4 yr. The leading two
terms amount to 13.25% and 11.06% respectively for a total of 24.3% of the
total lod variance.

Figure 5: Fourier spectrum of the lod time series (1962-2021).

Let us turn to the time series (1978-2022) of coordinates of the pole of
rotation ($m_{1}$, $m_{2}$ ; see Lopes et al. (2022a)). The Fourier analysis
of the modulus m (hereafter PM, for polar motion) is shown in Figure 6. The
spectrum features only two sharp peaks at 1 yr and 1.2 yr. Components with
decadal to 80 yr (Gleissberg cycle) periods are present but out of the
picture, and so is the trend known as the Markowitz drift (Markowitz (1968),
Le Mouël et al. (2017, 2021)). The 1/3 yr and higher harmonics seen in NHSI,
SHSI, and lod are not found in PM The $\sim$1.2 yr component is actually a
doublet at 1.19 $\pm$ 0.00(4) and 1.20 $\pm$ 0.00(4) yr, known as the Chandler
wobble (Chandler (1891a, b)). The three components, Markowitz, Chandler and
annual, amount respectively to 7.5, 40.4 and 19.8% of the signal variance, for
a total of 67.7% (Lopes et al. 2021).

Figure 6: Fourier spectrum of the coordinates of the rotation pole ($m_{1}$,
$m_{2}$) time series (1978-2022)

Figures 7(a) and 7(b) compare the annual and semi-annual SSA components of PM
and lod. The annual components are slightly modulated ($<$10%) and are close
to pure sinusoids. As a result, they almost perfectly correlate, with a phase
difference of 41 $\pm$ 3 days between m2 and lod (i.e. quadrature; Figure 5).

| DSE | NHSI | SHSI
---|---|---|---
lod | 152.5 $\pm$ 4.1 | 33.0 $\pm$ 1.1 | 153.7 $\pm$ 7.1
$m_{1}$ | 65.6 $\pm$ 2.1 | 153.8 $\pm$ 4.2 | 11.5 $\pm$ 2.4
Table 1: Phase differences of the annual lines of the pairs given as column
and line headings (in days, that is in this case very close to 1 day = 1
degree).

(a) Superimposition of the SSA annual components of the $m_{2}$ and lod time
series (top: 1978-2022; bottom enlargement: 2010-2022)

(b) SSA semi-annual component of the lod time series (2010-2022).

Figure 7: Annual and semi-annual components extracted from lod and PM

The prominence of an “astronomical” annual component in these time series of
course makes one relate it to the revolution period of Earth around the Sun.
We compare the annual variations of the Sun-Earth distance DSE (in au) with m1
(Figure 8, bottom) and lod (Figure 8, top). The phase lags are respectively
65.6 $\pm$ 2.1 days for DSE and $m_{1}$ and 152.5 $\pm$ 4.1 days for DSE and
lod.

Figure 8: Variations in the Earth to Sun distance DSE (in au; green) compared
to the first (annual) SSA component of $m_{1}$ (bottom) and lod (top). In the
middle curve, lod is offset by 152.5 days

##  5 Comparison of SSA annual and semi-annual components of polar motion and
length of day vs variations in sea-ice extent

We now compare the annual components of sea-ice extent NHSI (Figure 9) and
SHSI (Figure 10) with those of m1 and lod, in particular their phases (1). In
the case of the Northern hemisphere (Arctic), variations in sea-ice extent are
almost in phase with lod and in phase opposition with $m_{1}$. The phase
difference is 153.8 $\pm$ 4.2 days for m1 and 33.0 $\pm$ 1.1 days for lod (see
Figure 9 middle enlargement). NHSI is more strongly modulated (by 20% over 40
years, corresponding mainly to the trend) than lod. In the case of the
Southern hemisphere (Antarctic), variations in sea-ice extent are almost in
phase with m1 and in phase opposition with lod.

The semi-annual component of lod is compared with that of NHSI in Figure 9
(middle) and with that of SHSI in Figure 11 (top). The respective phase
differences are 51.8 $\pm$ 1.9 days (N) and 19.2 $\pm$ 1.5 days (S). Recall
that the semi-annual component is present in lod but not in $m_{1}$. Thanks to
the Euler-Liouville system (Laplace (1799)), we can integrate the lod to
obtain the theoretical polar motion that should accompany it. This is done in
Figure 11 (bottom). The agreement in phase is excellent.

Figure 9: Comparison of SSA annual components of pole motion $m_{1}$ (bottom;
in arc seconds) and lod (top and middle enlargement; in ms) with that of NHSI
(Arctic sea-ice; in 106 km2). Top = 1978-2022; middle and bottom = 2010-2022.

As shown in the previous section, these annual and semi-annual components of
NHSI and SHSI are much larger (up to 20 times) than the trends observed since
1978. The mechanisms generally suggested as a forcing factor of the trends
(global warming, winds, see Zwally et al. (2002); Zhang (2007); Turner et al.
(2015); Parkinson (2019)) must be dwarfed by a first order geophysical or
astronomical forcing. Of course, the annual oscillating components of NHSI and
SHSI are in opposite phases in both hemispheres (Figure 3(a) and 4(a)). SHSI
is not significantly modulated whereas NHSI is. This is reminiscent of the
classical behavior of a forced oscillator. Polar motion indeed possesses a
forced annual component that is generally attributed to climate and
ocean/atmosphere sources (e.g. Jeffreys (1916); Lambeck (2005) chapter 7).

Figure 10:  Comparison of SSA annual components of pole motion $m_{1}$
(bottom; in arc seconds) and lod (top and middle enlargement; in ms) with that
of SHSI (Antarctic sea-ice; in 106 km2). Top = 1978-2022; middle and bottom =
2010-2022.

Figure 11: Comparison of SSA semi-annual components of pole motion $m_{1}$
(bottom; in arc seconds) and lod (top and middle; in ms) with those of NHSI
(middle and bottom; Arctic sea-ice; in 106 km2) and of SHSI (top; Antarctic
sea-ice; in 106 km2) for 2010-2022. $m_{1}$ in the lower frame is not observed
but is computed by integrating lod (see text).

The lack of modulation of the largest (SHSI) forced component suggests an
alternate mechanism. If we follow Laplace’s paradigm, the torques exerted by
the Moon, Sun and planets play the leading role (if not all) as the source of
forcing (modulation) of many (if not most) geophysical phenomena (generalized
Earth tides). These forces (and torques) lead to changes in the inclination of
the Earth’s rotation axis, transferring stresses to the Earth’s solid and
fluid envelopes, setting Earth masses and resulting in thermal dissipation.
This description is essentially that of Milankovic (1920) on paleo-climates
and ice ages: more than variations in eccentricity, it is variations in
inclination of the rotation axis that lead to the large annual component of
melting and re-freezing of sea-ice.

We have seen in Figure 8 the excellent correlations of lod and $m_{1}$ with
the Sun-Earth distance DSE (respectively in phase opposition and in phase. It
is noteworthy that the phase differences of the annual components of NHSI with
$m_{1}$, SHSI with lod and DSE with lod are all equal to 153 days, which
happens to be half of the Euler period of 306 days (Table 1).

Let us follow Lambeck’s (1985, chapter 3, system 3.2.9) formulation of the
Liouville-Euler equations (see also Lopes et al. (2021)):

$\displaystyle\dfrac{i}{\sigma_{r}}(\dfrac{d\textbf{m}}{dt})+\textbf{m}$
$\displaystyle=\textbf{f}$ (1) $\displaystyle\dfrac{dm_{3}}{dt}$
$\displaystyle=f_{3}\qquad(\textbf{lod})$

$i=\sqrt{-1}$; $\textbf{m}=m_{1}+i*m_{2}$ (polar coordinate). This is a linear
first-order differential system. The annual oscillation of the forcing
function $\textbf{f}=f_{1}+i*f_{2}$ that follows sea-ice extent variations
should behave as $m_{1}$ or $m_{2}$, without phase changes. This is indeed
what is seen in Figures 9,10,11 where phases remain constant. The fact that
one recovers the phase differences of polar motion and lod in the leading
annual components of sea-ice strengthens previous findings (e.g. Courtillot et
al. (2021); Lopes et al. (2021, 2022a)). lod and $m_{1}$, $m_{2}$ are strongly
linked.

##  6 Discussion and Conclusion

The differences between the frequency ranges of length of day (lod) and polar
motion (PM) can be understood in the frame of Laplace (1799) ’s formulation of
the Liouville-Euler system, which consists of an equation for polar axis
inclination $\theta$ and one for the derivative of declination,
$\dfrac{d\psi}{dt}$, both depending on $\theta$ only. The second equation that
involves a derivative operator amounts to a high-pass filter333The derivative
operator is the reason why lod has a richer content in higher frequency
components than $m_{2}$., whereas the first behaves as an integrator, i.e.
passes lower frequencies.

Let us explore further consequences of Laplace’s formulation. We have seen
that there is no semi-annual component in polar motion (e.g. Lopes et al.
(2021, 2022a). According to the theory of forced oscillators (e.g. Landau &
Lifshitz (1989), the forcing frequency, here the annual oscillation, is at the
boundary between the two frequency domains. Figure 9 (center) shows that the
semi-annual components of NHSI and lod are almost in quadrature (51.8 $\pm$
1.9 days, close to 90/2 = 45 days for quadrature). Since there is a derivative
operator between lod and PM and since from Figures 7, 8 and 9 the forced
annual oscillations may be assumed to follow variations in the Earth to Sun
distance DSE, in lod, in PM and finally in NHSI and SHSI with remarkably
constant phase differences, then any semi-annual oscillation of lod should
theoretically also be present in PM. This would perturb polar motion and thus
must be eliminated (dissipated), which is why we find it in NHSI and SHSI, and
in quadrature with lod (Figure 9, bottom).

Integrating the semi-annual component of lod should give access to the same
semi-annual component of $m_{1}$ that does not exist but should be in phase
with the semi-annual component of NHSI, given the behavior of the ice series
annual component. The result is shown in Figure 9 (bottom) under the label
“calculated $m_{1}$”. The phases and modulations of the two curves are the
same (yielding in addition a prediction of the semi-annual component of NHSI).
This is why all forced oscillations and their harmonics are found in
variations in sea-ice, with very small amplitude modulations and remarkably
stable phase differences between both hemispheres and with lod and PM.

The generally accepted origin of the annual oscillation in PM is the
reorganization of fluid masses at the Earth surface. We may quote somewhat
extensively the introduction of Lambeck’s (2005) chapter 7:

“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. These discrepancies may be a
consequence of (i) inadequate data for evaluating the known excitations
functions, (ii) the neglect of additional excitation functions, (iii)
systematic errors in the astronomical data, or (iv) year-to-year variability
in the annual excitation functions”.

In other words, Lambeck (2005) explains that the annual oscillation of PM is
in part caused by variations in sea-ice and in part by variations of the
atmospheric circulation. Regardless of whether one follows Laplace (1799) or
Jeffreys (1916), the observations do show remarkably constant phase
differences. Yet, there is no reason for the periodic components of all
climate indices to have the same relative or absolute phases (e.g. Le Mouël et
al. (2019b)) (e.g. Figures 12(a) and 12(b)).

(a) AO Index, its mean (red) and spectrum (lower curve).

(b) AAO Index, its mean (red) and spectrum (lower curve).

Figure 12: AO and AAO Indices

It is common to find in the literature suggestions that the evolution of sea-
ice extent should be captured by climate indices, in the present case first
and foremost the Arctic oscillation AO (cf. Figure 12(a), Thompson & Wallace
(1998)) and the Antarctic Oscillation AAO (cf. Figure 12(b), Thompson &
Wallace (1998)). The AO is characteristic of exchanges in atmospheric mass
between the Arctic Ocean and the zonal ring centered on 45°N. Rigor et al.
(2002) have for instance shown how changes in surface winds associated with
the trends and fluctuations of AO influence sea-ice motion in the center of
the Arctic, and how these changes in ice motion in turn influence the
thickness and concentration of sea-ice, and the distribution of surface
temperature above the Arctic (Stroeve et al. 2007; Serreze & Meier 2019; Cohen
et al. 2020). The AAO monitors the periodic reinforcement and weakening of the
circumpolar vortex; it is also called the austral annular mode AAM. It is a
stationary mode, unlike the Antarctic circumpolar waves (White & Peterson
1996) that explain a large part of Antarctic climate variability on pluri-
annual scales. The two AO and AAO indices are widely used in order to explain
the evolution of sea ice since 1978. Yet, their spectra (Figures 12(a) and
12(b)) are devoid of any significant annual and semi-annual periodicities,
when the NHSI and SHSI series display annual and semi-annual periodicities
that amount to more than 98% of the total signal variance. Nor are annual and
semi-annual periodicities found in 10 MJO indices and several other index
series, such as PDO and ENSO (Le Mouël et al. 2019b). On the other hand, polar
motion and length of day are global phenomena that could explain the
regularity of phase and amplitude modulations of the sea-ice oscillating
components at 1 and 0.5 year.

Thus, the formulation by Laplace and the modern analysis of periodic and semi-
periodic components of sea-ice variations are not that far from one another,
although the direction of causality may have to be changed. If we follow
Laplace’s view, variations in sea-ice extent are a consequence of polar motion
as can be read from the Liouville-Euler equations. On the other hand, in the
modern interpretation, they are consequences of wind speed variations, that
should be reflected by indices AAO and AO.

The results obtained in this paper on phases and amplitudes of SSA components
of NHSI and SHSI allow one to prefer Laplace’s formulation. This formulation
allows one to understand the presence of the harmonic series (1, 1/2 , 1/3,
1/4, 1/5, 1/6, 1/7, 1/8 yr) in Figure 2(a). In a schematic way, variations in
sea-ice extent are the infinitesimal (incremental) expression of oceanic
motions, a topic that requires use of the theory of fluid mechanics of
turbulent flow. There is still no solution of the problem in the spherical
case (Schrauf 1986; Mamun & Tuckerman 1995; Nakabayashi & Tsuchida 1995;
Hollerbach et al. 2006; Mahloul et al. 2016; Garcia et al. 2019; Mannix &
Mestel 2021). But there is one in the case of a flow between two coaxial
cylinders (Taylor 1923; Landau & Lifshitz 1959; Chandrasekhar 1961; Frish
1995). The perturbation in flow velocity $v_{1}$ can be written as:

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

where integer $n$ is the order of symmetry ($n$=0 for axial symmetry) and $k$
is the wavelength of the instability. The acceptable values of the annual
oscillation are obtained by solving equation 2 under the appropriate boundary
conditions ($v_{1}$ = 0 for $r$ = $R_{1}$ and $r$ = $R_{2}$). The Reynolds
number $Re$ can be taken as $\dfrac{\Omega_{1}R_{1}^{2}}{\nu}$ (or
$\dfrac{\Omega_{2}R_{2}^{2}}{\nu}$ since $\dfrac{R_{1}}{R_{2}}$ and
$\dfrac{\Omega_{1}}{\Omega_{2}}$ are given). For fixed $n$ and $k$ the values
of $\omega$ form a discrete suite $\omega_{n}(k)$. By analogy with Earth, the
solid surface and free surface of the atmosphere are such that $v_{1}=0$, and
the only solutions for abs($\omega$) are 1 day and 1 year. We should therefore
find harmonics of these two values. A detailed study of the solution of
equation 02 in the case of co-axial cylinders is given by Chandrasekhar
(1961). The $v_{1}$ flows are stationary toric vortices also known as Taylor
vortices, regularly placed along the cylinder generatrices. In the case of two
cylinders rotating in the same sense, one finds two vortices rotating in
opposite directions, $2\pi/k_{cr}$, $k_{cr}$ being the critical wavelength
above which $v_{1}$ is not 0. So, we now know how to describe the figure of
the turbulent flow between two co-axial cylinders.

Given the above, it is no surprise to find the series of harmonics
$\omega_{n}(k)$ of the annual forcing. Also, the term $n\varphi$ in equation 2
forces the spatial symmetries of the flow. This is already known regarding the
variations of the sea-ice dipole in Antarctica (cf. Holland et al. (2017);
Bertler et al. (2018); Le Mouël et al. (2021b) and the variations of the
trends of global atmospheric pressure (cf. (Lopes et al. 2022b)). These space-
time variations seem to be better explained by Taylor-Couette type flow forced
by variations in pole motion (rotation), basically as envisioned by Laplace
(1799).

## Appendix A

Laplace (1799, vol. 5, cap. 1, page 347) in french:

“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: ces vents soufflent entre les tropiques, d’occident en
orient, et leur action continuelle sur la mer, sur les continents et les
montagnes qu’ils rencontrent, paraît devoir affaiblir insensiblement ce
mouvement de rotation. Mais le principe de conservation des aires, nous montre
que l’effet total de l’atmosphère sur ce mouvement doit être insensible; car
la chaleur solaire dilatant également l’air dans tous les sens, elle ne doit
point altérer la somme des aires décrites par les rayons vecteurs de chaque
molécule de la Terre et de l’atmosphère, et multipliées respectivement par
leur molécules correspondantes; ce qui exige que le mouvement de rotation ne
soit point diminué. 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 aux tremblements de Terre, et en général, 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 B

Terrestrial reference frame. $m_{1}$ and $m_{2}$ are the coordinates of the
rotation pole. $\psi$ and $\theta$ are the declination and inclination
introduced by Laplace (1799).

Laplace gives the full equations with the Sun and Moon included:

$\theta=h+\dfrac{3m}{4n}.(\dfrac{2C-A-B}{C})\left\\{\begin{array}[]{lll}&\dfrac{1}{2}.\sin(\theta).[\cos(2\nu)+\dfrac{\lambda
m}{m^{\prime}}\cos(2\nu^{\prime})]\\\
&-(1+\lambda).m.\cos(\theta).\Sigma\dfrac{c}{f}.\cos(ft+\varsigma)\\\
&+\dfrac{\lambda
c^{\prime}}{f^{\prime}}.\cos(\theta).\cos(f^{\prime}t+\varsigma^{\prime})\end{array}\right.$
(3)

$\begin{array}[]{ll}\dfrac{d\psi}{dt}&=\dfrac{3m}{4n}.(\dfrac{2C-A-B}{C})*\\\
&\left\\{\begin{array}[]{lll}&(1+\lambda).m.\cos(\theta)-\dfrac{\cos(\theta)}{2}.\dfrac{d}{dt}[\sin(2\nu)+\dfrac{\lambda
m}{m^{\prime}}.\sin(2\nu^{\prime})]\\\
&(1+\lambda).m.\dfrac{\cos^{2}(\theta)-\sin^{2}(\theta)}{\sin(\theta)}.\Sigma
c.\cos(ft+\varsigma)\\\
&\lambda.m\dfrac{\cos^{2}(\theta)-\sin^{2}(\theta)}{\sin(\theta)}.c^{\prime}.\cos(f^{\prime}t+\varsigma^{\prime})\end{array}\right.\end{array}$

Note that $\theta$ and $\dfrac{d\psi}{dt}$ depend on $\theta$ but not $\psi$ .

The inclination $\theta$ of the rotation axis has the current value h in the
first equation. $\dfrac{d\psi}{dt}$ is linked to the Earth’s rotation,
therefore to the lod. On the right side of the two equations are the
ephemerids and masses of the Moon and Sun that enter the classical theory of
gravitation (see Appendix A in Lopes et al. (2021) for more details). Length
of day and polar inclination are clearly connected by the first relationship.
Thus, Laplace reduces the problem to a system of two equations for the
inclination and time derivative of the declination of the Earth’s rotation
axis. $\theta$ and $\dfrac{d\psi}{dt}$ (and the norm that can be considered as
a known constant) give the direction of the polar rotation axis and its
variations. The time difference (in ms) between the theoretical and measured
Earth rotation is proportional $\dfrac{\psi}{v}$, $v$ being the rotation
velocity (and the Earth’s radius is a constant). Either $\psi$ alone, or $v$
alone, or both can vary. We assume the former, since the mean rotation rate
apparently remains constant, as was already noted above, and the two equation
imply studying the time derivative of declination of the rotation axis, thus
studying a quantity that is linearly related to the derivative of lod.

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11institutetext: University of Maryland, Baltimore County, USA
22institutetext: University of Arizona, USA
33institutetext: Leibniz University Hannover, Germany
44institutetext: McGill University, Canada 5University of Stavanger, Norway
6Guangdong Pharmaceutical University, China
7University of Southern Queensland, Australia
44email: 1{xinh1,wenbinzhang}<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
# LSTM Based Sentiment Analysis for Cryptocurrency Prediction

Xin Huang 11 Wenbin Zhang 11 Yiyi Huang 22 Xuejiao Tang 33 Mingli Zhang 44
Jayachander Surbiryala 55 Vasileios Iosifidis 33 Zhen Liu 66 Ji Zhang 77

###### Abstract

Recent studies in big data analytics and natural language processing develop
automatic techniques in analyzing sentiment in the social media information.
In addition, the growing user base of social media and the high volume of
posts also provide valuable sentiment information to predict the price
fluctuation of the cryptocurrency. This research is directed to predicting the
volatile price movement of cryptocurrency by analyzing the sentiment in social
media and finding the correlation between them. While previous work has been
developed to analyze sentiment in English social media posts, we propose a
method to identify the sentiment of the Chinese social media posts from the
most popular Chinese social media platform Sina-Weibo. We develop the pipeline
to capture Weibo posts, describe the creation of the crypto-specific sentiment
dictionary, and propose a long short-term memory (LSTM) based recurrent neural
network along with the historical cryptocurrency price movement to predict the
price trend for future time frames. The conducted experiments demonstrate the
proposed approach outperforms the state of the art auto regressive based model
by 18.5% in precision and 15.4% in recall.

##  1 Introduction

Since the birth of Bitcoin, there has been an enormous rise and interest in
the cryptocurrency, a decentralized digital asset developed by the blockchain
technology. This digital currency draws a lot of attention due to its
volatility which provides the opportunity for digital trading with high
return. The total market capitalization of cryptocurrencies has increased from
1 billion dollars to 400 billion dollars in the past decade, with the number
still increasing [1, 2, 3].

On the other hand, artificial intelligence are major drivers of technological
change [4, 5, 6, 7], while the emergence of social media such as Twitter,
Reddit and Facebook also makes the latest news and social media posts about
financial markets widely accessible. Investors have therefore been utilizing
such a variety of digital resources to make trading decisions. Previous
studies discovered evidence of such correlation between stock price movement
and social media [8, 9, 10]. Sentiment on cryptocurrency social media content
with negative emotions, e.g., fear and sadness, neutral emotions, e.g., calm
and not sure, or positive emotions, e.g., trust and happiness, can be used to
predict cryptocurrency price fluctuations and further to assist the investment
decision making. This paper focuses on this trending theme, proposing a
recurrent neural network with long short-term memory (LSTM) by utilizing the
sentiment analysis of social media to predict the real time price movement of
the digital currency.

##  2 Related Work

Over the past decades, a variety of machine learning techniques have been
developed to predict the price movement for the stock market using social
media, such as opinion analysis of twitter feeds [11, 12, 13, 14]. [8] used
neural networks and daily Twitter feeds as extra predictors to forecast the
daily up and down changes in the closing values of the Dow Jones Industrial
Average. Moreover, [15] found the Long Short-Term Memory (LSTM) combined with
a Twitter sentiment analysis outperforms other machine learning models such as
Support Vector Machine in predicting the stock price.

Recent studies have also successfully applied sentiment analysis in various
applications, such as predicting the movie revenues [16], analysing sentiment
towards US presidential candidates in 2012 [17]. In the English sentiment
analysis, Valence Aware Dictionary and sEntiment Reasoner (VADER) [18] is used
to classify sentiment in tweets. In this paper, we designed a crypto sentiment
dictionary that is customized to cryptocurrency and Chinese Weibo posts. We
use LSTM [19] as the neural network learning layer and combine it with the
sentiment analysis method to develop a crypto sentiment analyzer that can
predict the price movement of cryptocurrency.

##  3 Methodology

Figure 1 shows the end to end architecture of the LSTM based sentiment deep
learning model in predicting the real time price fluctuations. We crawled user
posts from China’s most popular social media platforms Sina-Weibo, and created
a crypto-specific sentiment dictionary with domain-expert knowledge, and then
the LSTM recurrent neural network was used to model the sentiment information
and make real time prediction for the price trend.

Figure 1: Architecture of LSTM based cryptocurrency sentiment analysis and
price movement prediction. Crypto sentiment dictionary is created to generate
the crypto word embedding, LSTM is to learn the sentiment information, and the
majority voting on the output of LSTM sentiment analyzer is used to predict
the price going up or down.

###  3.1 Data Collection

Chinese investors exchange crypto information via news articles and social
media platforms, especially using Sina-Weibo, Wechat and QQ groups. We collect
a large-scale Weibo corpus from crawling Chinese microblogs on Sina-Weibo with
the cryptocurrency keyword, in particular, Bitcoin, ETH or XPR. The number of
crawled cryptocurrency tweets from Weibo is 24,000, as well as 70,000 comments
to them, from the most recent 8 days.

###  3.2 Crypto Sentiment Dictionary

The general sentiment dictionary created by natural language processing (NLP)
is not applicable in the crypto domain. We introduce a novel way to build a
crypto specific sentiment dictionary that can capture the unique
characteristics of the crypto social communities. Table 1 shows an example
dictionary that is particular to the Chinese crypto words in Sina-Weibo. The
first step of generating a crypto sentiment dictionary is to create the
vocabulary of the crypto words. We manually label the crawled Weibo posts with
ranking, and use seed sentiment words selected by crypto domain experts to do
bootstrapping, which adds high frequency new words in highly positive/negative
weibo posts into the crypto corpus.

After the crypto corpus is generated, we then create an index mapping
dictionary in such a way that the frequently occurring crypto words are
assigned lower indexes, similar to the traditional natural language
processing. Finally we generate a crypto word encoding for each individual
post and use that encoding vector as the training data for the RNN model in
sentiment analysis.

Table 1: Crypto specific sentiment dictionary building for Chinese words. Chinese | Informal Translation | Implied Sentiment
---|---|---
韭菜 | Bag Holder | Investors have profit loss
新高 | New High | Investors are excited
下车 | Abandon Ship | Investors rush to sell off

###  3.3 LSTM based Sentiment Analyzer

We develop a long short-term memory network (LSTM) based sentiment analyzer
for crypto social media posts. LSTM enables the network to learn long-term
relation, by utilizing forget and remember gates that allow the cell to decide
which information to block or transmit based on its strength and importance.

The social media post is first tokenized according to crypto word vocabulary
and fed into the embedding layer, which converts the word token into the
crypto word embedding. The LSTM based recurrent network is trained by taking
the sequence of the embedding feature vector. A fully connected layer is used
to transform the output of the LSTM later and activated with sigmoid to output
the prediction. The labels of the posts used in training were manually labeled
and encoded with positive (1), neutral (0) and negative (-1).

##  4 Evaluation

We used the most recent 7 days’ Sina-Weibo posts from top 100 crypto investors
accounts as training data and the next 1 day’s posts as testing. We use
Precision and Recall to measure the performance of our LSTM sentiment
predictor. Precision measures the model’s ability to return only relevant
instances and recall measures the model’s ability to classify all relevant
instances.

We compare our method with the time series auto regression (AR) approach [20]
to evaluate the performance. As Table 2 shows, our approach outperforms the AR
approach by 18.5% in precision and 15.4% in recall, exemplifies the
effectiveness of the LSTM in analyzing the sentiment of social media content.

Table 2: Precision and recall on evaluating LSTM sentiment analyzer and AR. Method | Precision | Recall
---|---|---
Auto Regression | 73.4% | 80.2%
LSTM Sentiment Analyzer | 87.0% | 92.5%

##  5 Conclusion

In this paper we introduce a crypto sentiment analyzer by utilizing the
recurrent neural network to model the social media sentiment. The model is
developed using LSTM and achieves higher precision and recall than the
traditional auto regressive approach. The current sentiment analyzer can be
used to predict the price fluctuation of the cryptocurrency and integrated to
an autonomous trading system to assist the buying or selling of digital
assets. Immediate future directions include extending these results in
conjunction with our previous works [21, 22, 23, 24, 25] for fair credit
reporting and allocation of health care resources as well as unbiased scene
graph generation.

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