# On the Annual and Semi-Annual Components of Variations in Extent of Arctic and Antarctic Sea-Ice

^{1}

^{2}

^{*}

## Abstract

**:**

_{1}, while the reverse holds for the 1-yr component of SHSI. The semi-annual component appears in the lod and not in ${m}_{1}$. The annual and semi-annual components of NHSI and SHSI are much larger than the trends, leading us to hypothesize that a geophysical or astronomical forcing might be preferable to the generally accepted forcing factors. The lack of modulation of the largest (SHSI) forced component does suggest an alternate mechanism. In Laplace’s theory of gravitation, the torques exerted by the Moon, Sun, and planets play the leading role as the source of forcing (modulation), leading to changes in the inclination of the Earth’s rotation axis and transferring stresses to the Earth’s envelopes. Laplace assumes that all masses on and in the Earth are set in motion by astronomical forces; more than variations in eccentricity, it is variations in the inclination of the rotation axis that lead to the large annual components of melting and re-freezing of sea-ice.

## 1. Introduction

## 2. Available Data on the Extent of Sea Ice in Both Hemispheres

## 3. SSA Analysis of NHSI and SHSI Data on Sea Ice Extent

**iSSA**), and now do the same for the sea ice extent. We refer the reader to these papers and to the Golyandina and Zhigljavsky’s book [37] for the SSA method, to [38] for the properties of the Hankel and Toeplitz matrices that it uses, and to [39] for the singular value decomposition algorithm SVD).

**(SVD)**

**X**with dimensions $L\times K$ takes the shape

**X**can then be represented as a simple linear sum of elementary matrices

**X**${}_{i}$. If all eigenvalues are equal to 1, then decomposition of

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

**X**($d=\mathrm{rank}\phantom{\rule{4pt}{0ex}}\mathbf{X}=max\left\{i\right|{\lambda}_{i}>0\}$), SVD allows us to write

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

**X**be a suite of L lagged parts (

**X**and ${\mathbf{X}}_{1}$,…, ${\mathbf{X}}_{K}$) of the linear basis of its eigenvectors. If we let

**X**${}_{I}$ that regroups indices I can then be written as

**X**,

**X**${}^{\left(1\right)}$, and

**X**${}^{\left(2\right)}$ be the respective embedding matrices of series $\chi $, ${\chi}^{\left(1\right)}$, and ${\chi}^{\left(2\right)}$. These two subseries are separable (even weakly) in Equation (5) if there is a collection of indices $\mathcal{I}\subset \{1,\dots ,d\}$ such that ${\mathbf{X}}^{\left(1\right)}={\sum}_{i\in \mathcal{I}}{\mathbf{X}}_{i}$, respectively, if there is a collection of indices such that ${\mathbf{X}}^{\left(1\right)}={\sum}_{i\notin \mathcal{I}}{\mathbf{X}}_{i}$.

**Y**provides us with

**iSSA**) approach from among many others. Because relation (5) is linear, we can iterate the decomposition. We start with a small value of L (looking for the longest period), which we then increase until we obtain a quasi-Hankel matrix (Steps 1 and 2). We then extract the corresponding lowest-frequency component that was subtracted from the original signal. We again increase 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 frequencies to high.

## 4. Complementary Results for Polar Motion and Length of Day

**m**(hereafter PM, for polar motion) is shown in Figure 4. 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, though out of the picture, as is the trend known as the Markowitz drift [40,41,42]. The 1/3 yr and higher harmonics seen in NHSI, SHSI, and lod are not found in PM The ∼1.2 yr component is actually a doublet at 1.19 ± 0.00(4) and 1.20 ± 0.00(4) yr, known as the Chandler wobble [43,44]. These three components, Markowitz, Chandler and annual, respectively amount to 7.5, 40.4, and 19.8% of the signal variance, for a total of 67.7% [2].

_{1}(Figure 6, bottom) and lod (Figure 6, top). The phase lags are respectively 65.6 ± 2.1 days for D${}_{SE}$ and ${m}_{1}$ and 152.5 ± 4.1 days for D${}_{SE}$ and lod.

## 5. Comparison of Annual and Semi-Annual SSA Components of Polar Motion and Length of Day vs. Variations in Sea Ice Extent

_{1}, and are in phase opposition with lod.

## 6. Discussion and Conclusions

“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”.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

“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

**Figure A1.**Terrestrial reference frame; ${m}_{1}$ and ${m}_{2}$ are the coordinates of the rotation pole, while $\psi $ and $\theta $ are the declination and inclination, respectively, introduced by Laplace [11].

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**Figure 1.**Temporal evolution of SHSI (red) and NHSI (blue): (

**a**) raw southern hemisphere (SHSI, red) and northern hemisphere (NHSI, blue) sea-ice data and (

**b**) Fourier spectra of the SHSI (red) and NHSI (blue) series in Figure 1a.

**Figure 2.**Annual and semi-annual components extracted from SHSI (red) and NHSI (blue): (

**a**) SSA annual component of SHSI (red) and NHSI (blue) from 1978 to 2022 (

**top**) and their phase difference (

**bottom**, in days); (

**b**) SSA semi-annual component of SHSI (red) and NHSI (blue) from 1978 to 2022 (

**top**), an enlargement of the period from 2010 to 2022 (

**middle**), and their phase difference (

**bottom**, in days).

**Figure 4.**Fourier spectrum of the coordinates of the rotation pole (${m}_{1}$, ${m}_{2}$) time series (1978–2022).

**Figure 5.**Annual and semi-annual components extracted from lod and PM: (

**a**) superimposition of the SSA annual components of the ${m}_{2}$ and lod time series (

**top**: 1978–2022;

**bottom**: enlargement of 2010–2022); (

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

**Figure 6.**Variations in the Earth-to-Sun distance D${}_{SE}$ (in au; purple) compared to the first (annual) SSA component of ${m}_{1}$ (

**bottom**) and the lod (

**top**). In the middle curve, lod is offset by 152.5 days.

**Figure 7.**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 ic; in 10${}^{6}$ km${}^{2}$): top = 1978–2022; middle and bottom = 2010–2022.

**Figure 8.**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 10${}^{6}$ km${}^{2}$): top = 1978–2022; middle and bottom = 2010–2022.

**Figure 9.**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 10${}^{6}$ km${}^{2}$) and SHSI (

**top**, Antarctic sea-ice in 10${}^{6}$ km${}^{2}$) for 2010–2022; ${m}_{1}$ in the lower frame is not observed, but is computed by integrating lod (see text).

**Figure 10.**AO and AAO Indices: (

**a**) AO Index with its mean (red) and spectrum (lower curve); (

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

**Table 1.**Phase differences of the annual lines of the pairs, shown as column and line headings (in days, which in this case is very close to 1 day = 1 degree).

D${}_{SE}$ | NHSI | SHSI | |
---|---|---|---|

lod | 152.5 ± 4.1 | 33.0 ± 1.1 | 153.7 ± 7.1 |

${m}_{1}$ | 65.6 ± 2.1 | 153.8 ± 4.2 | 11.5 ± 2.4 |

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**MDPI and ACS Style**

Lopes, F.; Courtillot, V.; Gibert, D.; Mouël, J.-L.L.
On the Annual and Semi-Annual Components of Variations in Extent of Arctic and Antarctic Sea-Ice. *Geosciences* **2023**, *13*, 21.
https://doi.org/10.3390/geosciences13010021

**AMA Style**

Lopes F, Courtillot V, Gibert D, Mouël J-LL.
On the Annual and Semi-Annual Components of Variations in Extent of Arctic and Antarctic Sea-Ice. *Geosciences*. 2023; 13(1):21.
https://doi.org/10.3390/geosciences13010021

**Chicago/Turabian Style**

Lopes, Fernando, Vincent Courtillot, Dominique Gibert, and Jean-Louis Le Mouël.
2023. "On the Annual and Semi-Annual Components of Variations in Extent of Arctic and Antarctic Sea-Ice" *Geosciences* 13, no. 1: 21.
https://doi.org/10.3390/geosciences13010021