# On the tilt of the Earth’s polar axis ($\kappa\lambda\iota\mu\alpha$): Some
’impressionist’ remarks

Courtillot V  Université Paris Cité, Institut de Physique du globe de Paris,
CNRS UMR 7154, Paris, France  Lopes F  Université Paris Cité, Institut de
Physique du globe de Paris, CNRS UMR 7154, Paris, France  CNRS UMR7196, INSERM
U1154, Museum National d’Histoire Naturelle, Paris, France  Kossobokov V
Institute of Earthquake Prediction Theory and Mathematical Geophysics, Moscow,
Russia  Zuddas P  Sorbonne Université, METIS, UMR 7619, 4 Place Jussieu,
75005, Paris, France  Gibert D  LGL-TPE - Laboratoire de Géologie de Lyon -
Terre, Planètes, Environnement, Lyon, France  Boulé J.-B  CNRS UMR7196, INSERM
U1154, Museum National d’Histoire Naturelle, Paris, France  Le Mouël J.-L
Université Paris Cité, Institut de Physique du globe de Paris, CNRS UMR 7154,
Paris, France

(August 28, 2024)

###### Abstract

In this lengthy letter, we wanted to discuss the concept of climate based on
definitions established for over a century and direct observations that we
have been collecting for more than a century as well. To do this, we present
and discuss the remarkably stable maps over time of the various physical
parameters that make up the climate corpus: solar temperature, atmospheric
pressure, winds, precipitation, temperature anomalies. This impressionistic
tableau that we are gradually sketching as our reflection unfolds leads us to
the following proposition: What if, as Laplace first proposed in 1799 and
later Milankovic in 1920, ground temperature were merely a consequence of
climate and not a separate parameter of climate in its own right?

##  1  Introduction

In ancient Greek, $\kappa\lambda\iota\mu\alpha$ referred to the inclination of
the ground and, on a larger scale, to the inclination of the Earth towards the
pole starting from the equator, hence the concept of climate ([1], p.1102). It
was well understood that there was a connection between the angle at which the
sun’s rays intersect a given point on the Earth’s surface (relative to the
local vertical) and the climate of the region in question. In his work, which
remains a reference to this day, Lamb ([2], page 5) defines climate as the
average of meteorological phenomena, taken over several decades. This average
is characterized by a trend, a standard deviation from this trend, as well as
extreme phenomena associated with the trend. Lamb ([2], page 4) lists these
meteorological phenomena: they include temperature, atmospheric pressure,
humidity, clouds, precipitation, sunshine, and this list is not exhaustive.

Each one of these phenomena can be considered individually. The question that
arises then is to determine the ’appropriate’ duration for this time average
in order to distinguish as efficiently as possible the domain of climate from
that of meteorology. A ’universal’ agreement was reached during the
1935-Warsaw and 1957-Washington international meteorological conferences (cf.
[3]). A duration of 30 years was chosen, based on the observation of a
’natural’ cycle lasting 30 years in atmospheric circulation (eg. [4, 5, 6,
7]), known as the Brückner-Egeson-Lockyer cycle (BEL cycle, cf. [8]).

In our lengthy letter, we will systematically address each of the physics that
define the climate: solar temperature (Section 2), variations in atmospheric
pressure (Section 3) linked to the Earth’s axis rotation. Next, we will
discuss gradient maps of this pressure (Section 4), which give rise to both
the circulation cells of atmospheric air masses (Hadley, Ferrel, polar, see
Section 5) and are, with the ground topography, associated with climate
indices (Section 6). We will then treat about precipitation maps (Section 7)
and conclude with arguably the most debated parameter, the temperature anomaly
maps (Sections 8 to 11). We will conclude our letter with an in-depth
discussion and a reevaluation of the theories proposed by Milanković (1920,
[9]) and Arrhenius (1896, [10]).

##  2  On temperature and the Sun

One of the parameters mentioned by Lamb ([2]) is temperature, that one might
first assume to have its origin in the flow of energy and matter from the Sun.
However, there is no obvious 30-year cycle in temperature, or to be more
precise, this cycle remains of relatively low amplitude (cf. [11, 12]). This
observation can be extended to solar cycles, such as the sunspot cycles (eg.
[13, 14]), that represent part of the Sun’s activity. Sunspots, with
observations that date back to the early 18th century, exhibit time variations
that, from a spectral analysis perspective, can be decomposed into several
quasi-periodic cycles, among which the 11-year component, known as the Schwabe
([15]) cycle, is the best known. Yet, the signature of this cycle is only
weakly present in global average temperature records (eg. [16, 17, 12, 18]),
whereas it is clearly (if paradoxically) visible, for example, in
dendrochronological analyses (eg. [19, 20, 21, 22]), that are widely used in
the study of past climates (eg. [23, 24, 25, 26]).

One might ask whether the significance of temperature in what is consider to
be climate should be reconsidered. Isn’t discussing variations in the
temperature of the Earth’s lower atmosphere (which all agree have a
significant portion originating from astronomy and astrophysics) while also
mentioning a ’solar constant’111The flow of energy from the Sun at the
distance of Earth’s orbit a contradiction? The solar ”constant” was first
estimated by Pouillet ([27]) to be 1230 W.m-2. It is now estimated to average
about 1366 W.m-2, with a cyclic variation of approximately 0.1 to 0.3% about
its mean over an 11-year period (eg. [28, 29, 30]).

In his Treatise on Celestial Mechanics, Laplace ([31], vol. 5, chap. 1, page
347) wrote: “We have shown ($\ldots$) that the mean rotational motion of the
Earth is uniform, assuming that this planet is entirely solid, and it has just
been demonstrated that the fluidity of the sea and the atmosphere should not
alter this result. The motions induced by the Sun’s heat in the atmosphere,
giving rise to the trade winds, would seem to diminish the Earth’s rotation:
these winds blow between the tropics from west to east, and their continuous
action on the sea, the continents, and the mountains they encounter appears to
gradually weaken this rotational movement. However, the principle of
conservation of areas shows us that the overall effect of the atmosphere on
this motion must be negligible; for the solar heat, by equally expanding the
air in all directions, should not alter the sum of the areas described by the
vector radii of each molecule of the Earth and the atmosphere, multiplied
respectively by their corresponding molecules. This requires that the
rotational motion not be diminished. Thus, we are assured that while the
analyzed winds reduce this motion, the other atmospheric movements that occur
beyond the tropics accelerate it by the same amount”.

In this passage, the French scientist clearly considers the effect of
variations in solar temperature, and even temperature in general, on the
movements of the Earth’s rotational pole to be negligible. This is in line
with Milanković ([9])’s theory and holds true regardless of the duration of
the observation period (eg. [32]). Laplace ([31]) and Lagrange ([33]) were
convinced that the movements of the mobile parts of our planet stem from the
overall motion of the Earth, which is influenced by astronomical phenomena.
Furthermore, they recognized the direct action of astronomical forces on the
oceans, with coastal tides being the primary example.

##  3  On Rotation, Pressure, Flow and Winds

The law of conservation of areas mentioned by Laplace ([31]) can be
interpreted as representing the balanced exchange of angular momentum through
friction at the boundary between the atmosphere and the continents. In this
specific case, the system of the ‘solid Earth and atmosphere’ can be
considered as a system of concentric rotating spheres. Indeed, according to
Laplace ([31]), the effect of seas and oceans on the Earth’s overall motion is
equivalent to the effect these same oceans would have if they were solid. The
flow resulting from such a stationary configuration is known as Taylor-Couette
flow (cf [34]). While a formal solution has long been available for concentric
cylinders (cf. [35], Chapter 27, pages 99 to 103), obtaining a formal
mathematical solution for concentric spheres remains challenging even today
(eg. [36]). The steady-state of this flow is characterized by symmetric
patterns about the axis of rotation (both for spheres or cylinders) known as
the Taylor rolls.

We recently had the opportunity to test this last assertion of Laplace ([31])
by examining in detail the space-time evolution of maps of the trend of sea-
level atmospheric pressure (SLP) since 1850 (cf. [37]), as well as the space-
time evolution of maps of seasonal oscillations extracted from SLP, also since
1850 (cf. [38]). In the first study, we observed the presence of a time-stable
Taylor-Couette pattern (consistent with flow solutions) around the rotational
poles in both the Northern and Southern Hemispheres, which we named
’Triskeles’. As for the second study, it allowed us to highlight symmetry
rolls around the axes of rotation of the poles and also revealed the existence
of a harmonic series of seasonal forcing in the observations (1 year, 1/2
year, 1/3 year, 1/4 year, and 1/5 year), which is another manifestation of a
Taylor-Couette flow solution.

These patterns are often considered to correspond to the existence of winds in
the locations associated with these patterns. In order to assess this, we
started with raw observations, namely, monthly maps of sea-level atmospheric
pressure (SLP) since 1850, with a resolution of 5∘by 5∘. These maps are
provided by the Met Office Center1, and their production is described by Allan
and Ansell  ([39]). In Figure 1, we present the mean value extracted from
these SLP maps: this mean value has fluctuated by less than 0.1% since 1850.

Figure 1: Map of mean sea-level pressure trends since 1850. Robinson
projection.

##  4  The Pressure Gradient

In Figure 2, we have calculated the gradient of the average sea-level
pressures and superimposed it on the map of the gradient of topography in
order to determine the location of the friction zones mentioned earlier.

As expected, there is a strong relationship between the pressure gradients and
the orientation and magnitude of the man mountain ranges. The pressure
gradients organize themselves into large structures, often oriented east-west
and bounded by topographic gradients, which are often oriented north-south.
Moving from west to east, one can clearly observe how pressure gradient
anomalies, both negative (blue) and positive (red), encounter the American
Cordillera. From north to south, this occurs for the Brooks and Alaska ranges,
then in Canada along the Yukon, and finally in America along the Rockies.
Similar arrangements are found along the Andes mountain range, from Guatemala
to southern Patagonia.

Figure 2: Map of gradients of mean sea level pressure superimposed on the map
of gradients of topography (black). Robinson projection.

Another example, which partly explains the mystery of the Asian monsoon (eg.
[40]), extends from India to Mongolia. Although weak in the Indian Ocean, the
pressure gradient anomaly in this region is positive and changes sign after
crossing the Himalayas. The boundary is particularly well marked by the
topography gradient. This significant role of topography is of course well-
known (eg. [2], pages 84 to 87).

##  5  Atmospheric Cells and Climate Change

The pressure gradient forms zonal bands, alternating in sign approximately
every 30∘of latitude, similar to the commonly accepted distribution of
atmospheric circulation cells: the Hadley ([41]) cell between the equator and
$\pm$30∘, the Ferrel ([42]) cell between latitudes $\pm$30∘and $\pm$60∘, and
the polar circulation cells beyond $\pm$60∘. The mechanism generally invoked
to explain these cells is based on thermal exchanges (cf. [2], Chapter 3). At
the equator, where the temperature is highest, air masses expand and rise to
the tropopause, at about 15 km altitude, where temperature reaches a minimum.
High-altitude air masses are pushed towards the tropics, gradually cool, and
initiate their descent about $\pm$30∘. At polar latitudes, downward convection
drives air masses toward the equator, that warm in the process. The combined
action of the Hadley and polar cells gives rise to the mid-latitude Ferrel
cells.

One has to take into account an observed increase in the anomaly of average
surface temperature across the globe (cf. [43], eg. [32, 18]), at least since
the early 1980s, primarily in the Northern Hemisphere. This increase, on the
order of a degree Celsius, results in the melting of Arctic sea ice, while the
area of Antarctic sea ice has been increasing (but to a lesser extent; eg.
[44, 45, 46, 47, 48]). This should lead to changes in the latitude and size
(location and dimensions) of the three types of cells. Although this issue has
been a subject of discussion (eg. [49, 50, 51, 52]), the recent warming does
not seem to be reflected in all observations, as evidenced by the stability of
sea-level atmospheric pressure trends since 1850 (cf. Figures 1 and 2)

##  6  Climate Indices

Climate indices have been developed to study the impact of oceans and the
atmosphere on Earth’s climate. These indices take the form of time series that
describe the behavior of specific regions of the oceans and the Earth’s
atmosphere (eg. [53, 54, 55]). Among the most commonly used indices are El
Niño, a warm coastal current off the coast of Peru, the PDO (Pacific Decadal
Oscillation), which describes surface temperature variations in the Pacific
Ocean basin (cf. [56]), and the SAM (Southern Annular Mode), which is
generally defined as the difference in mean zonal sea-level pressure between
latitudes 40∘S and 65∘S (cf. [57]).

All these indices, which are based on temperatures, marine pressures,
rainfall, as well as the cyclicity and nature of winds recorded at sea level,
can be accessed in a comprehensive list available on the NOAA website1. We
seek to highlight potential interconnections between these geophysical
variables, interconnections for which there may not necessarily exist a
physical model. The idea of subjecting these observations to multivariate
statistical analysis methods, such as PCA (Principal Component Analysis, cf.
[58]) or SSA (Singular Spectrum Analysis, cf. [59]), naturally comes to mind.
Both of these statistical methods benefit from the power of Singular Value
Decomposition (SVD, cf. [60]). They can be summarized as follows.

Let $\mathcal{D}$ be a data matrix; its singular value decomposition is
$\mathcal{D}=U\lambda V^{\star}$. The principal columns of matrix $U$
represent the trends of the entire dataset; the vectors in $U$ represent the
’directions of greatest variation’ in the dataset. The diagonal values of
$\lambda$ are comparable to the ‘energy’ or ‘representativeness’ of these
components. SVD allows the construction of an empirical model without the need
for an underlying theory, which becomes more accurate as more terms are
integrated into it (cf. [61], chapters 7.6 to 7.8).

In Figure 3, we have outlined the geographical areas where the main climate
indices are defined and we have superimposed them on the pressure gradient map
(see Figure 2). The first index, which is the most significant in terms of
geographical extent, is the South Annular Mode (SAM, cf. [57]). The second
index relates to the Pacific Decadal Oscillation (PDO, cf. [56]), followed in
the third position by El Niño-Southern Oscillation (ENSO, cf. [62]). In the
fourth position, we find the North Atlantic Oscillation (NAO, cf. [63]),
followed in the fifth position by the Atlantic Multidecadal Oscillation (AMO,
cf. [64]). In the sixth position, we have the Madden Julian Oscillation (MJO,
cf. [65]), followed by the Pacific North American Index (PNA, cf. [66]) in the
seventh position. In the eighth position, we selected the Eastern
Atlantic/Western Russia Index (ER/WR, cf. [67]), and finally, in the ninth
position, we chose to represent the Tropical Southern Atlantic Index (TSA, cf.
[68]).

Figure 3: Map of pressure and topography gradients (Figure 2), superimposed on
the areas where the main climate indices are defined: 1 = South Annular Mode;
2 = Pacific Decadal Oscillation; 3 = El Niño-Southern Oscillation; 4 = North
Atlantic Oscillation; 5 = Atlantic Multidecadal Oscillation; 6 = Madden Julian
Oscillation; 7 = Pacific North American Index; 8 = Eastern Atlantic/Western
Russia; 9 = Tropical Southern Atlantic.”

We could have continued with the forty climate indices that exist today, but
that would only have reinforced the main observation made in Figure 3. As
already pointed out, although several parameters are taken into account to
construct these climate indices, it is clear that sea-level atmospheric
pressure dominates all others. As we will see, this natural division of the
indices is not reflected in temperature anomaly maps.

To our knowledge, we may have been the first to detect and extract the full
series of quasi-periodic variations that had already been observed in sunspots
(cf. [54]) from the entire set of climate indices. Several studies have since
confirmed our findings (eg. [69, 70, 55, 71, 72]).

However, we wish to update these conclusions, with an important remark. All
climate indices appear to be closely connected by their geographical
distributions (Figure 3). Pressure anomalies are well defined over a period of
observation of just over 170 years. In a series of articles (cf. [73, 74,
75]), we have shown, based on Lagrange ’s mechanics ([33]) and Laplace ’s
treatise ([31]), that it was possible to find the numerical values of the
periods of revolution of the planets involved in the motion of Earth’s
rotation pole, due to a simple exchange of angular momentum between these
planets and Earth (cf. [9, 76]). Similarly, we have shown that the (famous)
11-year period can be found in several Earth-bound time series such as the
length of the day (eg. [77], chapter 5.2; [78]), in global volcanic eruption
records from 1700 to the present day (cf. [79]), or in the growth rate of tree
rings in an entire forest of Tibetan junipers (cf. [22]). This consistently
recurring period is present in the evolution of all phenomena that record
various aspects of what is commonly referred to as climate (cf. [80, 81, 82]).
This 11-year period, clearly present in sunspots (cf. [15, 14]), is
nevertheless absent from temperature curves (eg. [12, 18]).

##  7  Rainfall

Another parameter that is part of the list distinguishing meteorology from
climate is rainfall. There is a series of data in the form of high-resolution
monthly maps (0.5∘by 0.5∘, between latitudes 60∘S and 90∘N), covering the
period from 1948 to 2016. These observations are made available by the
Hydrology Laboratory at the University of
Southampton222https://hydrology.soton.ac.uk/data/pgf/. We present the average
monthly rainfall map since 1948 in Figure 4.

The first obvious observation is the presence of a zonal band of high
rainfall, located near the equator and almost continuous. Moving from east to
west, one can sequentially notice a continuous red band stretching from 180∘W
to Central America, a high rainfall area above the Amazon rain-forest,
characterized by a single rainfall climate index, namely the Northeast Brazil
Rainfall anomaly (eg. [83]). This band tends to disappear above the Atlantic.
A more intense rainfall zone then reappears along the coasts of Liberia and
Sierra Leone, extending over the African rain-forest, the southern Sahel, and
stretching to about 17∘S. This zone practically covers the entire extent of
the continent’s forest. Approaching Ethiopia and Somalia, this anomaly
disappears only to reappear in the Indian Ocean south of Asia, clearly present
above all the islands of Indonesia. Another interesting observation is that,
at least during the period from 1948 to 2016, no notable anomaly appears on
average over Asia, especially concerning the Asian monsoon. Thus, the
alternation between the wet and dry seasons mitigates the variations on this
rainfall map. Broadly speaking, and as expected, rainfall is sensitive to
pronounced variations in topography (eg. above Burma, the Himalayas, or the
Alps and the Andes; Figure 4). The driest areas (dark blue) are located on
either side of the high rainfall zones (red). The overall structure is not
symmetrical about the equator but shifted by about 10∘to 15∘, reflecting the
distribution of continents.

Figure 4: Mean monthly rainfall map across the globe.

The second observation concerns the apparent complementarity between the
rainfall map and the pressure gradient map (see Figure 2). To better
appreciate this complementarity, we extracted from Figure 4 the areas where
rainfall exceeded 150 mm per month, and we superimposed them (in grayscale) on
the pressure gradient map (Figure 5).

Figure 5: Overlay on the pressure gradient map (cf. Figure 2) of precipitation
exceeding 150 mm per month, in grayscale.

On this Figure 5, that does not appear to have been discussed in the
literature and that we consider to be an original result, we can see that the
heavy precipitations, that we have observed to be generally zonal and
equatorial, occur in regions where the pressure gradient is nearly zero or
close to zero, indicating areas with pressure extrema. Finally, we note that
the precipitation band in the Pacific between 180∘W and Central America is
bounded by the ENSO (El Niño Southern Oscillation) and PNA (Pacific-North
American) indices, while the precipitation in South Asia and the Pacific is
found north of the MJO (Madden-Julian Oscillation) index and is constrained to
the west of Indonesia by India and to the west by the PNA.

##  8  Temperature Anomaly Maps

Figure 6 shows the mean temperature anomaly map since 1850 with a spatial
resolution of 5∘by 5∘. We have overlaid it with topography gradients. The
temperature anomaly is negative or null in the oceans (except the North
Atlantic), Canada, South America, Central Asia, and Australia. Conversely, it
is positive along the ocean-continent boundary from Alaska to Chile, in
Central Africa, the Himalayas, and at high latitudes, especially in the Arctic
region. Like the previous figure, it appears that the distribution of
temperature anomaly patterns does not follow a pattern similar to those of
winds, pressures, pressure gradients, or precipitation. For example, the
presence of a large temperature anomaly dipole, with a positive zone over the
United States and a negative zone over Canada, does not correspond to anything
in our other maps. Similarly, the Eastern Atlantic/Western Russia climate
index (cf. Figure 3, number 08) shows no correlation with the temperature
anomaly (cf. Figure 6).

Figure 6: Map of mean temperature anomalies since 1850.

While data related to temperature anomalies are regularly discussed and even
criticized (eg. [84]), it’s important to keep in mind that they do not
constitute direct measurements. These maps are constructed from actual
measurements but undergo various filtering and homogenization processes. For
more detailed information regarding these different treatments, especially
concerning the HadCrut data in Figure 6, we refer the reader to the works of
Rayner et Osborn (cf. [85, 86, 87, 88, 89]).

##  9  Data and Measurements. Variability of HadCrut Trends

The meaning of the terms ’data’ and ’measurement’ needs to be clearly
distinguished. Data originate from a sensor, typically a physical one, and are
associated with a margin of error. A measurement is the mathematical distance
that separates a piece of data from a theoretical model or equation. An
observation is simply that, a simple observation. It is understood that,
regardless of technological advancements in data acquisition or observational
instruments, what was estimated in the past retains its value. On the other
hand, a measurement depends on a theory or a model. If these evolve over time,
for whichever reason, it is obvious that the measurements, that are previously
calculated quantities, will need to be recalculated and thus modified.

Figure 7: Overlay of the 5 sets of HadCrut data Figure 8: Overlay of the 5
trends determined by SSA (from [32]).

This is illustrated by Figure 7 (from [32], Figure 01), where the 5 HadCrut
datasets are overlaid on the same graph. These datasets cover the following
time periods, in chronological order: 1870-2000 (HadCrutv, [85]), 1856-2006
(HadCrut2, [86]), 1850-2014 (HadCrut3, [87]), 1850-2021 (HadCrut4, [88]), and
finally 1850-2023 (HadCrut5, [89]). The different sets of measurements do not
strictly overlap in the past; however, they remain fairly close. Nevertheless,
these differences can have significant implications for forecasting. We have
addressed this issue, both for trends and annual oscillations (in [32]), and
revisit now some key points. If one examines closely the curves for HadCrut2
(purple), HadCrut3 (yellow), and HadCrut4 (red), the end of each time series
exhibits a plateau at different dates and durations (Figure 07b). This plateau
has led to discussions of what has come to be known as the ”climate hiatus”
(eg. [90, 91]). In particular, the HadCrut3 curve (yellow) shows a stable
plateau from 1990 to 2010. Many professional observers might have been tempted
to conclude that the temperature anomaly curve had stopped growing during that
time. However, one year later, with the introduction of the HadCrut4 curve
(red), this plateau had disappeared (Figure 8).

##  10  Temperature Anomalies as a Function of Latitude

The mean temperature anomaly curve (cf. Figure 7), like all average curves,
conceals subtleties and disparities. Similar to what we have done for previous
climate variables, we divided the anomaly map into six latitude bands of
30∘width each. For each of these bands, we assessed the evolution of the
average temperature over time using the most recent HadCrut series, version 5
(blue in Figure 8). The temperature anomaly curves for each latitude band are
shown in Figure 9.

A careful examination of the HadCrut (v5) average data, broken down into six
latitude bands, leads to two important observations. First, there are
significant differences between the two hemispheres. In the polar Southern
Hemisphere (top-left curve), the temperature anomaly has remained relatively
stable since 1850, staying within a range of $\pm$1∘C. However, in the polar
Northern Hemisphere (top-right curve), there is a half-oscillation with an
amplitude of +2∘C before 1960, followed by a modest linear increase of
approximately +2∘C333The following description is a preliminary draft and is
not the only possible one (see Courtillot et al. [7] and Le Mouël et al.
[12]). It helps to establish the order of magnitude of temperature differences
and their durations. However, over the past thirty years, the signal-to-noise
ratio has shifted in favor of noise, to the point when ’peaks’ exceeding +4∘C
appear quite regularly. These peaks can be observed globally in the entire
Northern Hemisphere, and they are present on all three curves in the right
column at the same moments. This latter point is somewhat surprising because
these ‘peaks’ are generally attributed to more or less significant variations
associated with El Niño (eg. [43], Chapter 2.4, page 370, and Paragraph 2.4.2,
page 371, regarding El Niño). The fact that temperature anomalies have
remained relatively constant above Antarctica since 1850, while they have
increased by 2 $\pm$ 2∘C above the Arctic since 1970, could explain why sea
ice is melting in the Northern Hemisphere while forming in the Southern
Hemisphere ([92, 45, 93, 47, 48]).

Figure 9: Temperature anomaly evolution curves by latitude band. At the top,
from left to right: between 90°S and 60°S, between 90°N and 60°N. In the
middle, from left to right: between 60∘S and 30∘S, between 60∘N and 30∘N. At
the bottom, from left to right: between 30∘S and 0∘, between 30∘N and 0∘.
Vertical dashed lines indicate a time reference (1960)

In the 30∘(N/S) - 60∘(N/S) latitude bands (see the middle curves in Figure 9,
the behaviors are still very different. In the Southern Hemisphere (left
curve), one could estimate that one has emerged from ”natural variability”
around the 1990s. The subsequent growth is approximately linear. Between 1990
and the Present, the temperature anomaly would have risen from +0.1∘C to
+0.5∘C, an increase of +0.4∘C. During the same period, in the Northern
Hemisphere, the temperature anomaly would have risen from about 0∘C to +1.5∘C,
which is three times larger.

This contrast between the two hemispheres raises questions about the physical
meaning of the notion of mean temperature and the ’average’ greenhouse effect.
If we crudely average the variations in temperature anomalies since 1990, we
observe that in the Southern Hemisphere, between 90∘S and 30∘S, the anomaly
has increased on average by about 0.2∘C ( (0∘C + 0.4∘C)/2 ), whereas in the
Northern Hemisphere, it has increased by about 2∘C ( (2∘C + 2∘C)/2 ), which is
ten times larger. ’Climate change’ primarily affects the Northern Hemisphere (
$>$ 60∘N).

The values around the equator, shown in the bottom curves of Figure 9, are
very similar. Since 1990, the temperature anomaly has increased by only 0.5∘C.

##  11  Temperature Anomalies as Functions of Time

In seeking to provide a physical interpretation for the temperature mean
trends, one can propose the following: the Earth’s lower atmosphere is
gradually warming from the North Pole to the South Pole. In Figure 9, we have
depicted the evolution of the map of average temperature anomalies in 20-year
intervals since 1850. The white areas, prominently visible in the first two
maps at the top left, as well as above Antarctica in the first five maps,
indicate the absence of measurements. It is worth noting that the number of
longitude-latitude pairs for which we have temperature measurements only
covered about 50% of the Earth’s surface in 1850, decreased to 40% after 1860,
then increased in steps to reach around 70% by 1960. It was only with the
advent of satellites that the coverage finally reached 100% of the Earth’s
surface (eg. [18], Figure 01, top curve).

Figure 10: Evolution of the median map of temperature anomalies. Top row, from
left to right: between 1850 and 1870, between 1850 and 1890, between 1850 and
1910. Middle row, from left to right: between 1850 and 1930, between 1850 and
1950, between 1850 and 1970. Bottom row, from left to right: between 1850 and
1990, between 1850 and 2010, between 1850 and 2023. The color code remains the
same for all 9 maps, linearly ranging from -1∘C (blue) to +1∘C (red).

In this Figure 10, we have retained the color scale, ranging from $\pm$ 1∘C,
which is consistent with the temperature range in the 30∘S-90∘S band. It is
clear, when looking at these nine maps, that indeed the Earth is warming, and
it has been doing so southward from the Arctic.

##  12  Discussion

Modern definitions of climate and meteorology were established during two
international meteorological conferences. Climatologists and meteorologists
agreed on a separation of 30 years, corresponding to the BEL cycle ([8]). They
did not take into account sea levels in their definition, but things have
changed significantly (eg. [2] or [43], Chapter 3, page 426). Douglas ([94])
begins his article with the following sentence: ’It is well established that
sea level trends obtained from tide gauge records shorter than about 50-60
years are corrupted by interdecadal sea level variation’. There appears to be
a contradiction between the duration of the BEL cycle and the one mentioned by
Douglas ([94]), which is twice as long. Therefore, it is possible that
conducting climate studies over a slightly longer period than 30 years and
attempting to estimate sea level since 1992 using satellites may lead to
confusion, potentially conflating a 60-year cycle with exponential growth (cf.
[95]).

The maps of mean sea-level pressures, their gradients, winds, and
precipitation (cf. Figures 1 to 5) show that these phenomena exhibit
remarkably stable patterns, both in time and space. Furthermore, these
patterns do not seem to inherently require the consideration of temperature
variations to explain their nature. This is a well-established but
unfortunately often overlooked fact in modern times, as we clearly illustrate
on our maps: sea-level pressure variations, their gradients, as well as what
climatologists call climate indices (eg. [54]), are primarily determined by
the gradients of terrestrial topography (cf. Figures 3 and 5) and are thus
connected by the same general physics. One can follow and verify Laplace ’s
reasoning ([31]). One can also recall the words of Fourier ([96], page 108):
’The intensity and distribution of heat on the surface of these bodies result
from the distance to the Sun, the inclination of the axis of rotation to the
orbit, and the state of the surface’.

The patterns we have highlighted, which are remarkably stable, are not found
in the map of surface temperatures (cf. [97]), nor even in those of
temperature anomalies (cf. Figures 6 and 10). It appears that temperature and
its evolution are largely unrelated to other physical aspects of the climate.

Two paradigms regarding the nature of temperature on Earth have long been in
opposition. The variations over time in the temperature at the Earth’s surface
can be attributed either to the Milanković ([9]) cycles or to the greenhouse
effect ([96, 10]). It is accepted that significant temperature and climate
variations observed during glacial cycles ($\sim$ 20,000 years, 400,000 years,
etc.) are linked to variations in the obliquity, precession, and eccentricity
of the Earth’s axis of rotation in our solar system (eg. [98, 99, 100, 9], as
detailed in [101]). For shorter periods, less than a hundred years, the
greenhouse effect is invoked, particularly variations in the concentration in
the atmosphere. Regarding this, one can read in Arrhenius ([10]), page 270:
’The world’s present production of coal reaches in round numbers 500 millions
of tons per annum, or 1 ton per km2 of the earth’s surface. Transformed into
carbonic acid, this quantity would correspond to about a thousandth part of
the carbonic acid in the atmosphere’.

It is surprising to note that when it comes to studying temperature variations
over less than two centuries, the Milanković theory is rarely, if ever,
invoked. This means that, in the perspective of modern climatologists, one
transitions instantly from a warm period to an ice age, and then to another
warm period. In Lopes et al. [32], we extended Milanković ([9])’s theory to
the past 250 years. Milanković ’s general relationship ([9], page 15, equation
20) relates the temporal variation (derivative) of the amount of insolation
received at the Earth’s surface to the variation in the inverse of the Earth-
Sun distance squared. This relationship leads to a perfect correspondence (in
the sense of Keogh [102, 103, 104]) between the HadCrut5 trend and the
variations in the Earth’s distance since 1850 (see Figure 11).

In this paradigm, it seems that Laplace ([31]) was the first to address the
question of the Sun’s action on Earth. The Sun is constant, at least for the
past 200 years during which we have measured its radiation. This radiation
puts the molecules of Earth’s atmosphere in motion through simple
thermodynamic effects. However, the issue is that, as highlighted by Laplace
([31]) (as well as [105, 33, 106, 96]), the cause of the rotations of the
solid Earth, oceans, and atmosphere is mainly found in celestial mechanics.
Thus, whether solar radiation is constant or not, exchanges of angular
momentum are constantly occurring between the planets themselves (eg. [76]),
between the oceans and our solid Earth (Laplace [31], shows that oceans behave
like solids in rotation at the first order), and between the atmosphere and
the solid Earth. However, the mass of Earth’s atmosphere ($5.29*10^{18}$ kg)
is negligible compared to that of the oceans ($1.35*10^{21}$ kg) and even more
so compared to that of the entire planet ($5.98*10^{24}$ kg).

In these conditions, as the atmosphere cannot exchange (strong enough)
momentum with the solid Earth, its effect becomes insensitive to the rotation
of the latter, and thus the molecules in motion due to solar radiation
compensate for their velocities: ’We are therefore assured that as the
analyzed winds decrease this motion, the other atmospheric movements occurring
beyond the tropics accelerate it by the same amount’. Thus, we lose the
spatiotemporal correlation between ground temperature or its variations
(independently of the cause) and other climate physics. This is precisely what
all the maps we have presented teach us. Therefore, this minimizes the
influence of the atmosphere and its physico-chemical properties, as
illustrated by the success of Milanković’s mathematical theory of climate, a
theory in which there is no atmosphere.

Figure 11: At the top (in blue and green), the inverse square of the
variations in the distance from the Earth’s rotation pole to the Sun during
the Summer and Winter solstices. In red, the time derivative of the
temperature anomaly trend extracted from HadCrut5. At the bottom, the same
curves corrected for a phase shift (15 years = 60 years/4). Figure extracted
from [32], Figure 11.

This raises questions about the importance and significance of the greenhouse
effect and albedo. Both of these effects appear to be of secondary importance
in our view. To support this idea, which is already evident in the data (see
all our maps), we begin with Arrhenius’s seminal article from 1896. In this
article, the author hopes that his theory, which is based on the effect of on
temperature, can explain ice ages and paleo-climates. He compares his
hypothesis to those existing at his time, drawing on a synthesis article by de
De Marchi ([107]), in which various topics are addressed, including (on page
273) changes in the position of the equinoxes, variations in the obliquity of
Earth’s axis of rotation, shifts in the pole positions, and Milanković ’s
mathematical theory of climates. Arrhenius ([10]) refutes all of de De Marchi
’s ([107]) hypotheses, including Milanković ’ ([9]).

Arrhenius ’s theory is based on a generalization of the blackbody radiation
law, known as Stefan’s law (see page 255). He attempts to extend this law to
account for the complexity of the Earth’s atmosphere, resulting in a complex
equation that relates various parameters such as albedo and surface
temperature (equation 03, page 256). To develop this equation, the author had
to use a large number of constants related to various components of the Earth,
biological, chemical, and physical reactions, and so on. Today, we know that
most of these ”constants” do not exist. Therefore, Arrhenius ’s conclusions
from 1896 can no longer be accepted as they are. Furthermore, Arrhenius
himself attenuates his conclusions from page 254 onward, including in the
synthesis by de Marchi. For example, he writes (pages 254-255): ’This
reasoning holds good if the part of the earth’s surface considered does not
alter its albedo as a consequence of the altered temperature. In that case
entirely different circumstances enter. If, for instance, an element of the
surface which is not now snow-covered, in consequence of falling temperature
becomes clothed with snow, we must in the last formula not only alter but also
’. As the author himself acknowledges, his reasoning is valid only if ‘the
earth’s surface considered does not alter its albedo’. However, volcanoes, the
biosphere, soil alterations can all modify the albedo.

The findings presented in this slightly ‘impressionist’ note confirm the
remarkable theoretical works of the Laplace ([31, 105, 33, 106]). Firstly,
temperatures are the only variables that do not follow the patterns of the
main climate parameters over time and space (see Figures 5 to 6). Secondly,
Earth has been warming over the past century from the North Pole towards the
equator (see Figures 10 and 11). The Earth’s axis of rotation has been tilting
since 1900 (eg. [75]), dragging the magnetic field1 with it (see [108]),
triggering volcanic eruptions (see [79]), influencing forest life rhythms (see
[22]), and even driving sea level changes (eg. . [109]).

Finally, could it turn out that temperature is neither a forcing factor of
climate nor a consequence of the main mechanisms that govern it ? Isn’t
Milanković ([9]) closer to the truth than Arrhenius ([10]) ?

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