hhmmsstime

# Did atmospheric thermal tides cause a daylength locking
in the Precambrian?
A review on recent results.

Jacques Laskar  Mohammad Farhat  IMCCE, CNRS, Observatoire de Paris, PSL
University, Sorbonne Université,
77 Av. Denfert-Rochereau 75014, Paris, France  Margriet L. Lantink  Department
of Geoscience, University of Wisconsin–Madison, Madison, WI 53706, Unites
States.  Pierre Auclair-Desrotour  IMCCE, CNRS, Observatoire de Paris, PSL
University, Sorbonne Université,
77 Av. Denfert-Rochereau 75014, Paris, France  Gwenaël Boué  IMCCE, CNRS,
Observatoire de Paris, PSL University, Sorbonne Université,
77 Av. Denfert-Rochereau 75014, Paris, France  Matthias Sinnesael  IMCCE,
CNRS, Observatoire de Paris, PSL University, Sorbonne Université,
77 Av. Denfert-Rochereau 75014, Paris, France

###### Abstract

After the initial suggestion by Zahnle and Walker (1987) that the torque
accelerating the spin rate of the Earth and produced by the heating of the
atmosphere by the Sun could counteract the braking lunir-solar gravitational
torque in the Precambrian, several authors have recently revisited this
hypothesis. In these studies, it is argued that the geological evidences of
the past spin state of the Earth play in favor of this atmospheric tidal
locking of the length of the day (LOD). In the present review of the recent
literature, we show that the drawn conclusions depend crucially on the
consideration of the stromatolite geological LOD estimates obtained by
Pannella at 1.88 and 2.0 Ga, which are subject to large uncertainties. When
only the most robust cyclostatigraphic estimates of the LOD are retained, the
LOD locking hypothesis is not supported. Moreover, the consideration of the
published General Circulation Model numerical simulations and of new
analytical models for the thermal atmospheric tides suggest that the
atmospheric tidal resonance, which is the crucial ingredient for the LOD
locking in the Precambrian, was never of sufficiently large amplitude to allow
for this tidal LOD lock.

Key words: Milankovitch cycles – Thermal atmospheric tides – Earth-Moon
distance – Tidal friction – Precambrian Earth– LOD resonant locking.

0

######

  1. 1 Introduction
    1. 1.1 Atmospheric thermal tides
    2. 1.2 A possible lock of the length of the day
  2. 2 Geological archives for Precambrian LOD estimates
    1. 2.1 Stromatolites
    2. 2.2 Tidal rhythmites
    3. 2.3 Cyclostratigraphy
  3. 3 Discussion of the recently published results
    1. 3.1 Mitchell and Kirscher,  (2023)
    2. 3.2 Wu et al,  (2023a)
      1. 3.2.1 The modeled gravitational tides: artificial resonances?
      2. 3.2.2 Atmospheric thermal tides: model limitations
      3. 3.2.3 The asymmetry of the Lamb resonance
      4. 3.2.4 The temperature problem
    3. 3.3 Bao et al,  (2022)
    4. 3.4 Farhat et al,  (2023)
  4. 4 Conclusions

##  1 Introduction

Since the work of Georges Darwin, it is known that the body tides exerted by
the Sun and Moon on Earth slow down the spin of the Earth and make the Moon
recede away (Fig.1) (Darwin, , 1879; MacDonald, , 1964; Goldreich, , 1966;
Kaula, , 1964; Mignard, , 1979, 1980; Hut, , 1981; Touma and Wisdom, , 1994;
Néron De Surgy and Laskar, , 1997).

Figure 1: In Darwin model, the Moon produces a tidal bulge on the Earth, but
as the Earth is not totally elastic, this bulge is driven by the fast rotation
of the Earth slightly off from the Moon direction. From this results a braking
torque that slows down the spin of the Earth. By conservation of the angular
momentum in the Earth-Moon system, the Moon goes away.

More elaborate tidal models take into account the oceanic tides which also
slow down the rotation of the Earth and let the Moon go away (Webb, , 1982;
Green et al, , 2017; Tyler, , 2021; Daher et al, , 2021), but none of these
tidal models could fit both the present tidal recession of the Moon of
$3.83\pm 0.008$ cm/yr (Williams and Boggs, , 2016) and the age of the Moon of
$4.425\pm 0.025$ Ga (Maurice et al, , 2020). Elaborated along the lines of
(Webb, , 1982), the recent semi-analytical model of (Farhat et al, , 2022)
provides a coherent scenario for the Earth-Moon tidal evolution, with an
excellent fit to the present recession rate and the age of the Moon. It starts
with a global ocean in the ancient eons, and then switches to a hemispheric
ocean model similar to (Webb, , 1982), but which follows the continental
evolution in the most recent times. Although no geological data was used for
the elaboration of the model, it is independently in good agreement with
geological estimates of the past Earth-Moon distance, and in particular of the
geological constraints obtained by cyclostratigraphic methods (Fig.2).
Nevertheless, following the original suggestion of (Zahnle and Walker, ,
1987), two recent papers propose that the spin of the Earth was trapped in a
resonance between the thermal atmospheric torque and the solid and oceanic
torque during the Precambrian Eon (Mitchell and Kirscher, , 2023; Wu et al, ,
2023a). Both studies advocate that this locking of Precambrian daylength is
supported by geological observations. However, such a scenario is incompatible
with the Earth-Moon evolution presented in (Farhat et al, , 2022). In this
review, which is aimed to the stratigraphic community, we will explicit the
differences in the modelling approaches and discuss the use of geological data
that can be compared with the models.

Figure 2: Evolution of the length of day (LOD) in the past following the model
of (Farhat et al, , 2022). The nominal LOD values are in purple, with the
uncertainty provided by the blue lines (adapted from fig.5 of (Farhat et al, ,
2022)). The circles represent the available cyclostratigraphic data with their
uncertainty, when available, from various sources: references within (Farhat
et al, , 2022) (in blue), (Zhou et al, , 2022) (in red), and (Zeeden et al, ,
2023) (in orange). Tidal rhythmites values are represented by yellow squares,
with references from (Farhat et al, , 2022) with the addition of a data point
at 3.2 Ga from the Moodies group (Eulenfeld and Heubeck, , 2023).

###  1.1 Atmospheric thermal tides

Atmospheric thermal tides have been recognized since the XVIII th century (see
Wilkes, , 1949). Due to the heating of the atmosphere by the Sun, the
atmosphere locally expands and the pressure decreases at the subsolar point,
which induces a redistribution of the mass of the atmosphere, with two main
components: a diurnal component, which at equilibrium is opposite to the
subsolar point, and a semi-diurnal component, orthogonal to the Sun direction
(Fig.3). As for the solid tides, as the Earth spin rotation is faster than its
orbital motion around the Sun, the Earth rotation drags these atmospheric
bulges with a positive offset from their equilibrium position. At present, the
gravitational attraction of the Sun on these bulges induces an accelerating
torque, opposite to the solid and ocean tides (Chapman and Lindzen, , 1970;
Goldreich and Soter, , 1966; Gold and Soter, , 1969; Ingersoll and
Dobrovolskis, , 1978; Dobrovolskis and Ingersoll, , 1980; Correia and Laskar,
, 2001, 2003; Correia et al, , 2003; Leconte et al, , 2015; Auclair-Desrotour
et al, , 2017, 2019).

Figure 3: Thermal atmospheric tides. Due to the heating of the Sun at the
subsolar point, a redistribution of the atmosphere creates two main
components: a diurnal component (in light blue), and a semi diurnal components
(in blue) that are offset from their equilibrium position because of the
Earth’s fast rotation. They create an accelerating torque on the Earth’s spin
motion.

###  1.2 A possible lock of the length of the day

Figure 4: LOD locking due to thermal atmospheric tides resonance for the
scenarios of (Zahnle and Walker, , 1987) (in red) and (Bartlett and Stevenson,
, 2016) (in grey). The geological indicators that are plotted are limited to
pre-2016 published results, adapted from (Laskar, , 2020) and (Williams, ,
2000) (see references therein). The dotted red line is the LOD provided by
equation 41 of (Laskar et al, , 2004) with a simple Darwin tidal model. The
dotted black line is an empirical fit using a simplified tidal model adjusted
to the geological data (Walker and Zahnle, , 1986). The stromatolite data
points at 1.88 and 2.0 Ga are from (Pannella, , 1972a, b).

At present, the thermal atmospheric tidal torque is a small part ($\sim 6.4$
%) of the solid and oceanic tidal friction (Volland, , 1990; Farhat et al, ,
2023), but there are two elements that can change this ratio, in favor of the
atmospheric thermal tides. First, the atmospheric torque amplitude is
dependent on the spin rate of the Earth, and as for the oceanic tides, there
exists a known resonance of the planetary Lamb wave (Bretherton, , 1969), that
we name here the Lamb resonance (Farhat et al, , 2023), occurring for a faster
spin value of the Earth, where the atmospheric torque is largely increased
(Lindzen and Blake, , 1972; Zahnle and Walker, , 1987; Bartlett and Stevenson,
, 2016). In addition, the oceanic tidal friction is at present close to a
resonance, but its value was smaller in the past, for a faster rotation spin
value (e.g. Farhat et al, , 2022). These elements led Zahnle and Walker,
(1987) to propose that at some time in the Precambrian, the accelerating
thermotidal torque couter-acted the braking luni-solar gravitational tidal
torque, which led to a lock of the length of the day (LOD) at about 21h for an
extended period of more than 1 Ga. This assumption was probably motivated by
the difficulty to fit the existing geological indicators of the past Earth-
Moon distance and LOD with more simple models (e.g. Williams, , 2000; Hinnov,
, 2018; Laskar, , 2020). This hypothesis was recently revisited by Bartlett
and Stevenson,  (2016).

Both in (Zahnle and Walker, , 1987) and (Bartlett and Stevenson, , 2016), a
large part of the observational constraints was provided from the Precambrian
estimates of the LOD resulting from stromatolites deposit analysis (Pannella,
, 1972a, b) (Fig.4).

##  2 Geological archives for Precambrian LOD estimates

In his review on geological archives for LOD estimates, Williams,  (2000)
mentioned several varieties of bio-archives (bivalves, corals, brachiopods)
which we will not consider here, as they were not present in the Precambrian.
For these, one can also consult (Rosenberg et al, , 1975) or (Lambeck, ,
1980). We will concentrate here on the stromatolites (2.1), the tidal
rhythmites (2.2), and the promising cyclostratigraphic records (2.3).

###  2.1 Stromatolites

The model of (Bartlett and Stevenson, , 2016) relies heavily on the adjustment
to stromatolite data (Pannella, , 1972a, b), although the authors themselves
warn the reader, and we can quote from (Bartlett and Stevenson, , 2016):
However, these data, particularly the early stromatolite data [Pannella,
1972], should not be taken too seriously. [Zahnle andWalker, 1987]
Paleontologists Scrutton [1978] and Hofmann [1973] also found these data to be
unreliable and unsuitable for precise quantitative analysis. These data have
also been used in a crucial manner in the recent studies (Mitchell and
Kirscher, , 2023; Wu et al, , 2023a; Bao et al, , 2022).

Stromatolites are layered organo-sedimentary structures formed by the capture
of sediments by microbial organisms, typically formed in shallow marine and
lacurtrine environments. They are some of the oldest known forms of life, and
are an important archive for studying Precambrian paleoenvironments. Studies
of recent analogues have shown that daily rhythms of biological growth and
binding of sediment can be preserved in stromatolites layering, as the
microorganisms (algae) respond actively to daylight (e.g. Logan et al, , 1964;
Monty, , 1967; Davies, , 1970; Gebelein, , 1969). In addition, environmental
fluctuations influence rates of growth and the supply of material, i.e.,
layering thickness, including tidal and seasonal variations (Gebelein and
Hoffman, , 1968; Pannella, , 1976; Pannella, , 1972b). As such, investigations
have been made of diurnal growth layers interpreted from fossil stromatolites
that can be structured in larger tidal and seasonal banding; these
observations have in turn been used to estimate past LOD and the length of the
lunar month.

The analyses of Precambrian stromatolites by Pannella,  (1972a, b) are a well-
known example of such a study, and, as discussed are often used in long-term
reconstructions of the past Earth-Moon system (e.g. Bartlett and Stevenson, ,
2016; Bao et al, , 2022; Mitchell and Kirscher, , 2023; Wu et al, , 2023a).
The three oldest, most critical LOD data that are often referenced are based
on stromatolite sequences from the ca. 1.88-Ga Gunflint Formation (Fralick et
al, , 2002), the correlative Biwabik Formation (Lake Superior region), and the
Paleoproterozoic Great Slave Supergroup (Northwestern Territories); the exact
stratigraphic origin and thus age of the latter sample is unknown, but is
usually placed around 2 Ga, following the compilation figure of Williams,
(2000). However, as emphasized in many studies (e.g. Hofmann, , 1973;
Scrutton, , 1978; Lambeck, , 1980), and by Pannella himself in the original
publication, these stromatolite-based estimates are rarely, if ever, suitable
for precise quantitative interpretation. The formation of daily laminae
depends on a fine environmental balance that can easily be disturbed, and
therefore stromatolite growth patterns are rarely considered to be complete
due to periods of non-deposition and/or post-depositional erosion by storms,
for example. For this reason, counts or sequences of laminae should generally
be interpreted as minimum estimates, and not as most likely values (Pannella,
, 1972a, b; Lambeck, , 1980). Another source of uncertainty arises from
ambiguities associated with determining the exact number of daily laminae per
lunar monthly or annual bundle, which is often characterized by a low
reproductibility rate, and further challenged by a lack of independent
temporal controls. This is a challenge that is, however, not unique to the
stromatolite archive (tidal rhythmites, for instance, feature similar
challenges). Concerning the stromatolite specimens studied by Pannella,
(1972a, b), we specifically note that his counts of the number of diurnal
laminae per larger seasonal growth band yield a significantly lower
interpreted number of days per year than the observed number of daily
increments between lesser growth marks that are believed to indicate the
synodic month, established from the same sequences. In addition, Mohr,  (1975)
arrived at a very different number and conflicting interpretation for time-
equivalent specimens. In conclusion, while stromatolites can provide useful
insights in past tidal dynamics, they should typically not be considered as
providing accurate or precise numerical values for past LOD reconstructions.

###  2.2 Tidal rhythmites

Tidal rhythmites are laminae deposits related to semi-diurnal or diurnal tidal
cycles that can occur in estuaries or deltas. Silty, and muddy sediment is
carried by ebb tidal currents. These currents transport the sediment in
suspension through the main ebb channel to deeper offshore water, where it
settles and forms graded layers. During slack water between tides, muddy caps
can be deposited on the sandy laminae. The amount of sediment carried by ebb
tidal currents and the effectiveness of tides in transporting and depositing
sediment are directly related to the tidal amplitude (Williams, , 2000). In an
ideal scenario, the analysis of the time series of the thickness of these
laminae should allow to recover all tidal periodicities:

  * •

The lunar day, interval between two passage of the Moon at the meridian.

  * •

The lunar synodic month, which separates two full Moons, and is thus recovered
by the recognition of spring tides of large amplitude, occurring at syzygy,
when the Moon and the Earth have same longitude, when seen from the Sun.

  * •

The tropical year, separating two passages of the Earth at the spring equinox
(also called vernal equinox), possibly determined as the time of maximal tidal
amplitude in the year.

  * •

Finally, if the record is sufficiently long, the nodal period of the Moon
(18.6 yr at present) could be recognized (Walker and Zahnle, , 1986).

Although very promising, this method has its drawbacks. The locations where
high-quality sequences of such laminae formations can be observed are rare.
Moreover, they often led to divergent analyses when the deposits are analysed
by different groups. The Weeli Wolli Formation in Western Australia, dated
2.45 Ga, was interpreted by (Walker and Zahnle, , 1986) on the basis of the
lunar nodal cycle and led to an Earth-Moon distance of $51.9\pm 3.3$ Earth
radius ($R_{E}$), while the analysis of (Williams, , 1989, 1990) led to the
much larger value of $54.6\pm 1.8\ R_{E}$, with the analysis of laminae
couplets, grouped in synodic fortnightly increments, and annual cycles.
Although the results of these two studies are still consistent within
uncertainty (given the very large error bars), these estimates are based on
two fundamentally different and mutually incompatible interpretations of the
same layering patterns. In the same way, the analysis of the Elatina Formation
in South Australia, dated 620 Ma, led Williams,  (1989, 1990) to determine an
Earth-Moon distance of $58.16\pm 0.30R_{E}$ while Sonett and Chan,  (1998)
derived $56.04\pm 0.03R_{E}$ from their analysis of the same sequence. Sonett
and Chan,  (1998) also re-analyzed their previous determination of the Earth-
Moon distance for the Big Cottonwood Formation in Utah, at 900 Ma, and found a
value corresponding to an Earth-Moon distance of $55.06\pm 1.44R_{E}$, while
their previous determination of the same sequence was $57.1R_{E}$ (Sonett et
al, , 1996). We note that the new determinations from (Sonett and Chan, ,
1998) are in agreement with the new model of (Farhat et al, , 2022) (Fig.2).
However, given the difficulties associated with interpreting Earth-Moon
parameters from these tidal sequences, additional independent studies should
be required to further verify the determination of (Sonett and Chan, , 1998).

###  2.3 Cyclostratigraphy

Due to the gravitational interactions of the Earth with the other planets, the
orbital plane of the Earth is moving in a complex motion composed of a slow
rotation (precession of the node) and a composition of nearly periodic motions
that make the inclination of the Earth orbital plane oscillate (e.g. Laskar, ,
2020). This induces as well an oscillation of the tilt of the Earth (or
obliquity, $\varepsilon$) of present amplitude 1.3 degrees around its averaged
value. The variation of insolation on the surface of the Earth depends also on
the precession of perihelion, and of the variation of eccentricity, which are
dominated by the so-called long eccentricity term of 405 kyr period and the
short eccentricities of main periods 95 kyr and 124 kyr. Finally, the pull of
the Moon and Sun on the equatorial bulge of the Earth induces a slow
precessional motion of its spin axis at $50.475838$ arcsec/yr that corresponds
to a period of about 26kyr (Laskar, , 2020). Neglecting the eccentricity of
the Earth and Lunar orbit, and the inclination of the Moon, the lunisolar
precession can be expressed as $\alpha\cos\varepsilon$, where $\varepsilon$ is
the obliquity and where the precession constant $\alpha$ is (eq.4.14 from
(Laskar, , 2020))

$\alpha={{\displaystyle 3}\over{\displaystyle 2}}G\,\left[{{\displaystyle
m_{\odot}}\over{\displaystyle a_{\odot}^{3}}}+{{\displaystyle
m_{M}}\over{\displaystyle\color[rgb]{1,0,0}a_{M}^{3}}}\right]{{\displaystyle{E\\!_{d}}_{0}}\over{\displaystyle\gamma_{0}^{2}}}{\color[rgb]{1,0,0}\gamma}$
(1)

where G is the gravitational constant, $\odot$ refers to the Sun, and $M$ to
the Moon, $m$ and $a$ are the masses and semi-major axes, $\gamma$ the Earth’s
spin angular velocity, $\gamma_{0}$ its present value, and ${E\\!_{d}}_{0}$
the dynamical ellipticity at present. As $a_{\odot}$ can be considered as
constant, the precession constant $\alpha$ thus depends mostly on the
evolution of $a_{M}$ and $\gamma$ which evolve in time under tidal dissipation
(highlighted in red in equation (1)).

The resulting changes in insolation drive climatic changes on Earth
(astronomical climate forcing) that can be recorded in the Earth’s sedimentary
archive. These sediments can today be studied (e.g. Gradstein et al, , 2004,
2012, 2020; Montenari, , 2018) and inform us on past astronomical changes.

Over very long timescales, beyond 60 Ma, the planetary orbital motions can no
longer be predicted with accuracy (Laskar et al, , 2011b, a), but for the
Earth-Moon evolution, the tidal dissipation will dominate, and a
reconstruction of the past evolution of the Earth-Moon distance can still be
achieved. The variation of $a_{M}$ and $\gamma$ will induce a change in the
precession period that can be imprinted in the sedimentary record (Berger et
al, , 1992; Meyers and Malinverno, , 2018). In a reverse way, the
determination of the precession frequency from the sedimentary record, and the
use of a dynamical model that will link the semi-major axis $a_{M}$ to the
angular spin velocity of the Earth $\gamma$ can allow to retrieve both $a_{M}$
and $\gamma$. This determination requires a time scale for the sedimentary
record, which can be provided either by absolute radiometric age dating, but
also, and more often by the use of the 405 kyr eccentricity period as a
metronome for stratigraphic cycles (see (Laskar, , 2020) and references
therein). In recent years, this technique for determining the past state of
the Earth-Moon system has made large progress, and many groups have obtained
converging results using various methods for the determination of the
precession frequency (e.g. Meyers and Malinverno, , 2018; Lantink et al, ,
2022; Zeeden et al, , 2023). Moreover, these data are in good agreement with
the tidal model of (Farhat et al, , 2022) (see fig.6 of (Farhat et al, , 2022)
and fig.5 of (Zeeden et al, , 2023)). Another crucial advantage of
cyclostratigraphy, relative to, for example, stromatolites and tidal
rhythmites, is the potential of independent age control by, for example,
radioisotopic geochronology and integrated stratigraphic approaches. Using an
integrated stratigraphic approach it is also possible to verify
interpretations in time-equivalent sections that should have the same time-
dependent astronomical signatures (e.g. Olsen et al, , 2019; Sinnesael et al,
, 2019). The coherence of these sets of data leads us to consider that they
are the most robust among the geological proxies for the determination of the
past precession frequency of the Earth and determination of Earth-Moon system
parameters.

##  3 Discussion of the recently published results

The solution of (Farhat et al, , 2022) is in agreement with the most recent
determinations of tidal rhythmites (Sonett and Chan, , 1998) and with the
recent cyclostratigraphic data (Meyers and Malinverno, , 2018; Lantink et al,
, 2022; Zhou et al, , 2022; Zeeden et al, , 2023) (Fig.2). One could thus
think that the fate of the thermal tides locking hypothesis was settled. But
the recent publication of the two papers (Mitchell and Kirscher, , 2023; Wu et
al, , 2023a) in major journals requires some additional discussion to clarify
the situation.

Figure 5: Comparison of the data and fit of (Mitchell and Kirscher, , 2023)
(black and red lines) with the model of (Farhat et al, , 2022) (purple curve
with uncertainty in blue). As in Fig.2, tidal rhythmites are yellow squares,
and cyclostratigraphic data are color circles (light blue with references in
(Farhat et al, , 2022), red from (Zhou et al, , 2022) and orange from (Zeeden
et al, , 2023)). The stromatolite data from (Pannella, , 1972a, b) are
highlighted with a dark green circle while the cyclostratigraphic data from
(Grotzinger, , 1986) is circled in light green. Adapted from Fig.2 of
(Mitchell and Kirscher, , 2023).

###  3.1 Mitchell and Kirscher,  (2023)

In their compilation of geological constraints of the Precambrian length of
the day, Mitchell and Kirscher,  (2023) have included most of the available
data. Their analysis is purely empirical. They search for the best linear fit,
made by pieces over sequences of data. One could wonder on the status of their
fit, which is not continuous, as a piecewise linear model would be. The goal
is thus not to find an empirical model for the Earth-Moon evolution, but to
search for the best fitted trends in the LOD over extended periods. From these
fits, they conclude to a probable lock of the LOD between 1 Ga and 2 Ga. When
comparing to the results of (Farhat et al, , 2022) (Fig.5), one can observe
that the stromatolites data at 1.88 Ga and 2 Ga from (Pannella, , 1972b, a)
are essential for the conclusions of (Mitchell and Kirscher, , 2023). If these
data, which are questionable as we discuss in section 2.1, are not taken into
account, the fit will no longer lead to this locked value of the LOD between 1
and 2 Ga.

It is also puzzling that the cyclostratigraphic point of (Grotzinger, , 1986)
is nearly exactly on the (Farhat et al, , 2022) curve (Fig.5). It should be
noted, however, that the datum point of (Grotzinger, , 1986) was not
originally given in that paper but was derived by Mitchell and Kirscher,
(2023). Grotzinger,  (1986) only proposed that there is eustatic sea-level
cyclicity within the Milankovitch frequency band recorded in platform
carbonates from the Rocknest Formation, at a scale of $1-15$ m and possibly of
$75-100$ m or $75-200$ m. (Mitchell and Kirscher, , 2023) then assumed that a
10 m cycle represents climatic precession and 87.5 m is related to short
eccentricity. However, this interpretation is poorly constrained. Due to the
large uncertainty of this data point, we should consider this as a simple
coincidence, until some new quantitative analysis in the spirit of (Meyers and
Malinverno, , 2018) is performed on the same 1.89 Ga Rocknest Formation sample
(Mitchell and Kirscher, , 2023).

Figure 6: Comparison of the data and fit of (Wu et al, , 2023a) (in black)
with the model of (Farhat et al, , 2022) (purple curve with uncertainty in
blue). As in Fig.2, tidal rhythmites are yellow squares, and
cyclostratigraphic data are color circles (light blue with references in
(Farhat et al, , 2022), red from (Zhou et al, , 2022) and orange from (Zeeden
et al, , 2023)). The stromatolite data from (Pannella, , 1972a, b) are
highlighted with a dark green circle. Note that these data points seem to be
misplaced in the (Wu et al, , 2023a) figure reproduced here (see note 1).
Adapted from Fig.3 of (Wu et al, , 2023a).

###  3.2 Wu et al,  (2023a)

In the recent work of Wu et al,  (2023a), the authors presented a new
analytical model of thermal tides to address the resonant locking hypothesis.
The model’s free parameters were constrained such that the resulting
thermotidal torque drives an LOD history that best fits their compilation of
LOD geological proxies. As previously, in figure 6, we compare (Wu et al, ,
2023a) (black curve) with (Farhat et al, , 2022) (purple curve). Here again,
one can see that the model of (Wu et al, , 2023a) relies heavily on the
stromatolite data of (Pannella, , 1972a, b) to establish the resonance locking
of the LOD111Note that these data points from Pannella (Pannella, , 1972a, b)
occur at a different age position in the figures $1-3$ of (Wu et al, , 2023a),
compare to previous LOD compilations (e.g. Williams, , 2000; Bartlett and
Stevenson, , 2016; Mitchell and Kirscher, , 2023). In particular, we note that
the two closely spaced points at ca. 1.63 Ga most likely represent Pannella’s
analyses of the time-correlative Gunflint (1972a) and Biwabik (1972b)
Formations dated at ca. 1.88 Ga (Fralick et al, , 2002). However, used
literature data were not provided in (Wu et al, , 2023a) to verify this
observation.. Moreover, (Wu et al, , 2023a) curve misses entirely the new
cyclostratigraphic determinations of the Earth-Moon state at 2.46 Ga obtained
by Lantink et al,  (2022) in the Joffre Gorge, Australia.

Wu et al,  (2023a) elaborated a physical model to support their claims.
Moreover, the authors performed a suite of GCM (General Circulation Model)
numerical simulations, using the LMD-G (Hourdin et al, , 2006) and PlaSim
(Fraedrich et al, , 2005) GCMs, to infer the Earth’s paleo-temperature
evolution that is required to generate the constrained history of the
thermotidal torque. We dedicate the rest of this section to discuss the
details behind the adopted model in (Wu et al, , 2023a) and its predictions.

####  3.2.1 The modeled gravitational tides: artificial resonances?

The dynamical evolution of the Earth’s rotational motion in (Wu et al, ,
2023a) is driven by the luni-solar gravitational tidal torque and the solar
thermotidal torque. For the former, the authors used the tidal history of
(Webb, , 1982), where Laplace’s Tidal Equations (the equations describing the
tidal response of a shallow fluid layer; LTEs hereafter) were solved semi-
analytically over a hemispherical equatorial ocean on the surface of the
Earth. While the work of Webb,  (1982) was seminal in coupling LTEs with the
dynamical evolution of the Earth-Moon system, the modeled history of the lunar
orbit in (Webb, , 1982) yielded a lunar formation epoch that is incompatible
with the geologically constrained lunar age (see Fig.3 of (Webb, , 1982)).

To efficiently remedy the latter discrepancy, Wu et al,  (2023a) tweak the
tidal dissipation history of Webb,  (1982) (see their Fig. S1) by normalizing
it with a constant factor, such that the resultant orbital history of the Moon
features its proper temporal origin. As a byproduct of this modeling choice,
the authors have modified the spectrum of oceanic normal modes in such a way
that tidal resonances are characterized with artificial amplitudes (see Green
et al, , 2017; Daher et al, , 2021; Farhat et al, , 2022).

Though the authors focus on modeling thermal tides, gravitational tides remain
the dominant driver of the Earth’s rotational evolution, providing the
background of the tidal torque upon which the thermotidal counterpart would
significantly contribute only in the vicinity of the Lamb resonance. As such,
since the authors are constraining the history of the total torque to fit a
compilation of geological LOD proxies, an artificial modeled spectrum of
gravitational tidal dissipation may yield an artificial spectrum of thermal
tides. Namely, the resultant thermotidal history could be characterized by
either an artificial timing of the Lamb resonance occurrence, an artificial
amplitude of the Lamb resonance, or both.

####  3.2.2 Atmospheric thermal tides: model limitations

For the thermotidal contribution to the Earth’s rotational history evolution,
Wu et al,  (2023a) develop a simplified analytic model of thermal tides that
is used to compute the thermotidal torque (Eqs. S28-S29 therein). The model is
parameterized by a number of free parameters (16 in total) that are
constrained such that the resulting thermotidal torque, added to the
gravitational tidal torque, would drive an LOD history that fits the
compilation of LOD geological proxies (see Figure 6).

The developed model essentially resembles a band-pass filter, similar to that
developed in Bartlett and Stevenson,  (2016). It ignores the Coriolis force,
which may be significant in the case of a fast rotator like the Earth, along
with the vertical velocity of tidal waves. The model also assumes an
isothermal structure of the atmosphere. This choice is classical in the
literature of atmospheric dynamics as it simplifies the mathematical framework
of the rather complex theory (e.g., Chapman and Lindzen, , 1970; Lindzen and
Blake, , 1972; Auclair-Desrotour et al, , 2019). However, for the Earth,
atmospheric temperature measurements (e.g., Figures 2.1-2.3 of Pierrehumbert,
, 2010) show that the massive troposphere (${\sim}80\%$ of atmospheric mass)
controlling the tidal mass redistribution is characterized by a negative
temperature gradient. The latter is in fact closer to an idealised adiabatic
profile than it is to an idealised isothermal profile. These modeling choices
could deliver inaccuracies in the determination of the resonant period (Farhat
et al, , 2023). However, this was somewhat compensated by the authors in
modeling the resonant period as a free parameter that is constrained by the
geological data.

The other essential quantity of interest is the amplitude of the thermotidal
torque when the resonance is encountered. The latter is dependent on several
variables, of which the least constrained in the case of the Earth is the rate
of energy dissipation by the atmosphere. Namely, as the atmosphere is heated
by the shortwave incident stellar flux and the infrared emission from the
ground, it dissipates energy via multiple pathways including radiative cooling
and frictional interactions with the surface. As it is difficult to properly
model these mechanisms in the analytical theory, energy dissipation is usually
modeled by a free parameter (the parameter $Q_{\rm th}$ in the work of Wu et
al, , 2023a). This unconstrained parameter predominantly controls the
amplitude and the spectral width of the resonant thermotidal torque and,
consequently, the lifetime of the Lamb resonance and whether it was sufficient
to counteract the gravitational tide.

Dissipative radiative transfer and atmospheric cooling, however, can be
properly accommodated in GCM simulations. To that end, Wu et al,  (2023a)
presents, to date, the first study that uses GCMs to simulate the Lamb
resonance specifically for the Earth. Their results, using the two
aforementioned GCMs, estimate the dissipation parameter to be $Q_{\rm
th}\approx 10$, which would render the maximum amplitude of the torque
insufficient for the LOD locking. For the LOD evolution, however, the authors
used values of $Q_{\rm th}$ that are one order of magnitude larger ($Q_{\rm
th}\approx 100)$ such that the thermotidal torque would be sufficient to
counteract the gravitational counterpart. Consequently, the used thermotidal
torque is amplified by a factor of ${\sim}30$ relative to its present
value222which means that the amplitude of the surface pressure anomaly is
amplified by a factor of ${\sim}60$ as can be inferred from Farhat et al,
(2023).. The author’s reasoning lies in the need for such a large thermotidal
torque so that the LOD proxies, specifically the stromatolites in (Pannella, ,
1972a, b), can be explained. This brings us back to Section 2.1 in questioning
the reliability of this data set as a robust constraint for informing
dynamical models, especially when present with evidence from GCMs to the
contrary.

Figure 7: The spectrum of the thermotidal torque as a function of the length
of day adapted from Fig.S4 of (Wu et al, , 2023a). The black dots are
simulated using the PlaSim GCM, while the solid black curve shows the
prediction of the analytical model of Wu et al,  (2023a). The red curve shows
a fit to the GCM results using the analytical model of Farhat et al,  (2023),
where the physical effects of the delayed thermal response of the ground were
taken into account. These effects proved to induce the notable amplitude
asymmetry between the peaks around the Lamb resonance. Namely, the
accelerating peak of the tidal torque is reduced in amplitude relative to the
decelerating peak.

####  3.2.3 The asymmetry of the Lamb resonance

An interesting signature of the GCM simulations of the Lamb resonance in (Wu
et al, , 2023a) lies in the spectrum of the thermotidal torque shown in their
Figure S4. We reproduce this spectrum in Figure 7. The GCM spectrum, shown by
the black dots, features an asymmetry in the peaks of the Lamb resonance
whereby the two peaks of the torque around the resonance do not share the same
amplitude. Namely, the accelerating part of the torque has an amplitude that
is almost half that of the decelerating part. The former part is required to
occur with a sufficient amplitude such that it counteracts the decelerating
gravitational torque, but it appears from this spectrum to be reduced. The
authors, however, ignored this signature present in the GCM simulations in
favor of the spectrally symmetric Lamb resonance obtained from their
analytical model, which is shown by the black curve in Figure 7.

In the recent work of Farhat et al,  (2023), the authors propose that such an
asymmetry can be obtained if one accounts for the thermal inertia budget in
the ground and the lowermost atmospheric layer. Namely, due to the thermal
inertia in these layers, the infrared heating of the atmosphere by the ground
becomes asynchronous with the incident solar flux. This delayed ground
response is shown to be responsible for maneuvering the atmospheric tidal
bulge in such a way that creates an amplitude asymmetry between the two peaks.
In Figure 7, we show by the red curve how the model of Farhat et al,  (2023)
can properly explain the spectral asymmetry of the GCM-produced spectrum when
taking into account the thermal inertia effects. It is important to note that
the reduction of the positive peak of the torque goes hand in hand with the
relative contribution of the ground in heating the atmosphere. Namely, the
more abundant the greenhouse gases are in the atmosphere, which is predicted
for the Precambrian from various geological proxies (see e.g., Catling and
Zahnle, , 2020), the more the atmosphere would be prone to infrared
thermotidal heating, and consequently the more the accelerating thermotidal
torque would be reduced.

####  3.2.4 The temperature problem

One naturally wonders how the discussed modeling limitations carry over to the
model predictions. The resulting timing of the Lamb resonance occurrence
requires a mean Earth temperature in the Proterozoic of $40-55^{\circ}$C,
computed by the authors using the PlaSim GCM (see Figure 7 of Wu et al, ,
2023a). Though a warm climatic interval fits evidence on a Proterozoic glacial
gap (e.g., Hoffman et al, , 2017), such extreme temperatures are in contrast
with geochemical analysis using phosphates (e.g., Blake et al, , 2010),
geological carbon cycle models (e.g., Sleep and Zahnle, , 2001; Krissansen-
Totton et al, , 2018), numerical results of 3D GCMs (e.g., Charnay et al, ,
2020), and the fact that solar luminosity was 10-25% lower during the
Precambrian (e.g., Gough, , 1981). Such extreme temperature estimates would
also require elevated amounts of partial pressure of CO2, reaching 200 mbar.
This exceeds inferred estimates from various geochemical proxies (see the
review by Catling and Zahnle, , 2020, and references therein). More
importantly, however, this temperature increase will enhance the asynchronous
thermotidal heating of the atmosphere by the ground in the infrared, as we
describe in Section 3.2.2. The latter would significantly attenuate the peak
of the tidal torque near resonance, rendering it insufficient for the LOD
locking. The adopted model in (Wu et al, , 2023a) did not account for this
feedback effect. Moreover, atmospheric dissipation is also enhanced with the
increased temperature, as discussed in (Farhat et al, , 2023), which has an
additional effect of attenuating the resonant amplitude of the torque.

###  3.3 Bao et al,  (2022)

Figure 8: Comparison of the data and fit of (Bao et al, , 2022) (in solid
blue) with the model of (Farhat et al, , 2022) (purple curve with uncertainty
in green). As in Fig.2, tidal rhythmites are yellow squares, and
cyclostratigraphic data are color circles (light blue with references in
(Farhat et al, , 2022), red from (Zhou et al, , 2022) and orange from (Zeeden
et al, , 2023)). The stromatolite data from (Pannella, , 1972a, b) are
highlighted with a dark green circle. Adapted from Fig.11 of (Bao et al, ,
2022).

While they do not invoke a thermal tides LOD trapping, Bao et al,  (2022) try
also to reconciliate the LOD history with the (Pannella, , 1972a, b) data.
This time, they propose that between 2 Ga and 1.5 Ga, a sudden growth of the
Earth core led to a reduction of the spin rate of the Earth. In figure 8, we
have compared their solution with (Farhat et al, , 2022) (purple curve). In
this case again, the scenario depends crucially on the stromatolite data of
(Pannella, , 1972a, b). If these data are removed, there is no longer the
necessity to search for some peculiar scenario, and it can be recognized that
the model of (Farhat et al, , 2022) fits most of the reliable geological data
in a satisfactory manner.

###  3.4 Farhat et al,  (2023)

Figure 9: Amplitude of the Lamb resonance with respect to the two parameters
$\sigma_{0}$ and $\alpha_{\mathrm{A}}$ of the (Farhat et al, , 2023) model. In
order for the thermotidal response to cancel the gravitational counterpart in
the Precambrian the parameters $(\sigma_{0},\alpha_{\mathrm{A}})$ need to be
below the solid black line. The dashed line defines the same threshold needed
for the Mesozoic, at 250 Ma. The horizontal shaded area corresponds to typical
values of the radiative cooling rate $\sigma_{0}$. The other shaded area
defines the region of parameter space corresponding to the presently observed
semi-diurnal tidal bulge. Adapted from (Farhat et al, , 2023).

In their recent work, Farhat et al,  (2023) have revisited the atmospheric
thermal tides computations for rocky planets and in particular for the Earth.
They have constructed an ab initio model of thermal tides on rocky planets
with a neutrally stratified atmosphere. This feature is a major change with
respect to previous models, where closed-form solutions are usually obtained
assuming that the atmosphere is isothermal (Lindzen and McKenzie, , 1967;
Chapman and Lindzen, , 1970; Lindzen and Blake, , 1972; Auclair-Desrotour et
al, , 2019; Wu et al, , 2023a). Although both atmospheric structures provide
appreciable mathematical simplifications, neutral stratification appears to
better capture the negative temperature gradient that characterises the
troposphere of the Earth, which contains most of the atmospheric mass. As the
stability of stratification with respect to convection determines the strength
of the Archimedean force exerted on fluid particles in the vertical direction,
the neutral stratification approximation annihilates the buoyancy effects in
the tidal response. The upward-travelling internal gravity waves are thus
filtered out from the solution, leaving only the horizontal compressibility
forces responsible for the propagation of the Lamb wave. Another major change
with respect to previous models (Lindzen and McKenzie, , 1967; Chapman and
Lindzen, , 1970; Lindzen and Blake, , 1972; Ingersoll and Dobrovolskis, ,
1978; Dobrovolskis and Ingersoll, , 1980; Auclair-Desrotour et al, , 2019; Wu
et al, , 2023a) is the consideration of heat absorption near the ground level
and heat exchange between the atmosphere and ground that takes into account
the thermal diffusive processes in the planetary surface layer. This model
allows to obtain a closed-form solution for the frequency-dependent
atmospheric tidal torque, and is in agreement with simulations using GCMs,
both for Earth-like and Venus-like planets. Specifically, when applied to the
Earth, their model predicts a resonant rotational period of 22.8 hr, which is
in agreement with a recent analysis of pressure data on global scales
(Sakazaki and Hamilton, , 2020)333This value is in agreement with the
$11.38\pm 0.16\,{\rm hr}$ semi-diurnal period obtained by analyzing the
spectrum of normal modes using pressure data on global scales (see Table 1 of
Sakazaki and Hamilton, , 2020, first symmetric gravity mode of wavenumber
$k=-2$), and the GCM prediction of Wu et al,  (2023b). As such, the model
predicts the occurrence of the Lamb resonance not in the Precambrian, but in
the Phanerozoic, with an amplitude that is insufficient to counteract the
luni-solar gravitational tidal torque. This does not exclude the occurrence of
the crossing of the resonance, but as the luni-solar gravitational tidal
torque remains larger than the thermotidal torque, no LOD trapping can occur.
The crossing of the resonance then results only in a small change of the
Earth’s rotational decceleration: the spin decceleration rate is slightly
increased before the resonance, and then reduced to roughly its previous value
after the crossing of the resonance.

The (Farhat et al, , 2023) model depends on two parameters
$(\sigma_{0},\alpha_{\mathrm{A}})$ , which are the cooling frequency and
opacity parameters, respectively. The frequency $\sigma_{0}$ is the inverse of
the timescale associated with energy dissipation, which is assumed to result
from radiative cooling in the model. The higher $\sigma_{0}$, the more
efficient is energy dissipation. The frequency $\sigma_{0}$ is thus tightly
associated to the amplitude of the Lamb resonance (thus related to the
parameter $Q_{\rm th}$ appearing in (Wu et al, , 2023a)). The opacity
parameter $\alpha_{\mathrm{A}}$ quantifies the fraction of incident Solar flux
that is transferred to the atmosphere in the thermal tidal forcing.
Consequently, this parameter takes its values between 0 (no tidal forcing) and
1 (maximal tidal forcing). Other model parameters are related to the present
day atmospheric gas mixture and surface temperature, and they are therefore
well constrained. For the thermotidal torque to cancel the gravitational
torque in the Precambrian (and thus the LOD locking to occur in the
Precambrian), the $(\sigma_{0},\alpha_{\mathrm{A}})$ pair needs to be below
the associated black solid curve of (Fig.9). Nevertheless, the observation of
the present thermal atmospheric response and the constraint on the cooling
frequency $\sigma_{0}$ deduced from the cooling timescale estimated by Leconte
et al,  (2015), impose to the $(\sigma_{0},\alpha_{\mathrm{A}})$ pair to be
inside the intersection of the two shaded regions, above the LOD lock
threshold. Moreover, (Farhat et al, , 2023) show that the crossing of the
resonance, within the limitations of their analytical model, most probably
occurred in the Mesozoic, and not in the Precambrian. In this case, the curve
to consider is the dashed line, which leads to an even less probable LOD
locking.

##  4 Conclusions

The famous astronomer Carl Sagan (1934-1996) used to say that extraordinary
claims require extraordinary evidences, which is another version of the
Occam’s razor in science, stating that simpler explanations should be
preferred to more complicated ones, in absence of strong arguments. Adapted to
the present problem of the evolution of the Earth-Moon distance and LOD, it
can be expressed as: Do we need a LOD lock by thermal tides to explain the
evolution of the Earth-Moon system over its age? Is there a strong evidence
for a LOD lock in the Precambrian?

The answer to the first question is clearly negative, as the (Farhat et al, ,
2022) model provides a coherent scenario for the tidal history of the Earth-
Moon system, without the need of a resonant atmospheric tidal lock.

We have seen also how all papers advocating for a LOD lock by thermal tides
(Bartlett and Stevenson, , 2016; Mitchell and Kirscher, , 2023; Wu et al, ,
2023a), or the alternate scenario of a growing Earth’s core (Bao et al, ,
2022) rely critically on the stromatolite LOD estimates of Pannella,  (1972a,
b).

However, as emphasied by Pannella himself, and by several authors that studied
these data afterwards, the validity of the stromatolite-based LOD estimates
derived from the Paleoproterozoic Gunflint-Biwabik Formations and Great Slave
Supergroup should be questioned (see section 2.1). There is thus at present no
reliable geological evidences to support these alternate scenarios. Moreover,
(Mitchell and Kirscher, , 2023) presented a cyclostratigraphy-based datum from
cyclicities in the 1.9-Ga Rocknest Formation (Grotzinger, , 1986) which is not
compatible with the stromatolite data of Pannella. In addition, the solution
of (Wu et al, , 2023a) complies with the questionable stromatolite data of
Pannella,  (1972b, a) but not with the more reliable cyclostratigraphic data
of (Lantink et al, , 2022).

The crucial importance of the stromatolite data at 1.88 Ga and 2.0 Ga used in
previous models is an important motivation for the search for alternate
estimates of the LOD in this time interval or more generally in the interval
of 1.5 Ga to 2.0 Ga. A preference should be given to high resolution
cyclostratigraphic data, in the spirit of (Meyers and Malinverno, , 2018). In
particular, it would be very useful to re-analyze the cyclostratigraphic data
at 1.9 Ga of (Grotzinger, , 1986). More generally, we would like here to
emphasize the importance of taking dating (age) uncertainty into consideration
when fitting variables through any type of empirical estimate derived from the
geological records.

The two recent analytical, semi-analytical, and numerical studies of Wu et al,
(2023a) and Farhat et al,  (2023), although providing opposite conclusions,
have improved our understanding of the possibility of atmospheric thermotidal
daylength locking in the Precambrian. The problem addressed in Wu et al,
(2023a) can be summarized as follows: two parameterized spectra of tidal
torques, one gravitational and one thermal, are combined, and the parameters
of the two counterparts are constrained such that the combined torque drives
an LOD evolution that fits a compilation of geological proxies. Much can be
appreciated in that work, especially in highlighting the significance of
Earth-Moon angular momentum depletion via thermal tides, simulating the Lamb
resonance for the Precambrian Earth using GCMs, and establishing, using GCMs,
for the first time a correlation between the resonant period, temperature
evolution, and atmospheric compositional variations. Moreover, in the limited
sense, the adopted analytical models of (Wu et al, , 2023a), laying down the
used spectra of the torques, appear to capture the fundamental dynamical
behavior of oceanic and atmospheric tides. However, a closer look at the
hierarchy of modeling assumptions in the two models and the constraints
imposed by the geological proxies reveal that the story is much more nuanced.
In short, stringent constraints on the LOD history were imposed by a subset of
quantitatively questionable proxies, as we discuss in Section 2.1. The latter
were combined with a spectrum of oceanic tides that does not physically
describe the tidal response of the Earth’s paleo-oceans. As such, the modeled
atmosphere was constrained to encounter the Lamb resonance with an unrealistic
amplitude for the torque and in a whistle-stop fashion such that the
stromatolite records in the Proterozoic can be explained.

Using a neutrally stratified analytical model that is more adapted to the
Earth’s atmosphere than the usual isothermal model, and taking into account
heat diffusion mechanism in the vicinity of the ground interface, Farhat et
al,  (2023) conclude that the amplitude of the Lamb resonance is not
sufficient for the thermotidal torque to counteract the luni-solar
gravitational tidal torque in the Precambrian. Moreover, their analysis
conclude that the crossing of the Lamb resonance most probably occurred in the
Mesozoic, and not during the Precambrian (see section 3.4), with even less
chance of daylength locking. Interestingly, the numerical GCM simulations of
Wu et al,  (2023a) allow to strengthen the analytical model of Farhat et al,
(2023) by probing the asymmetry of the Lamb resonance (Fig.7). These two
studies should provide the basis of future improved models for atmospheric
thermal tides, but for now, we should conclude, both by the consideration of
the geological evidences and by the comparison of theoretical models, that
there is no clear arguments allowing to state that LOD locking occurred in the
past history of the Earth.

Figures credits. Figure 3 from (Mitchell and Kirscher, , 2023), figures 3 and
S4 from (Wu et al, , 2023a), and figure 11 from (Bao et al, , 2022) were
adapted according to CC BY licence
(https://creativecommons.org/licenses/by/4.0/).

Acknowledgements. This project has been supported by the French Agence
Nationale de la Recherche (AstroMeso ANR-19-CE31-0002-01) and by the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation program (Advanced Grant AstroGeo-885250). MLL acknowledges funding
from the Heising-Simons grant no. 2021 – 2797.

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