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On the calculation of latitudinal insolation gradients throughout
the Holocene
Rodolfo G. Cionco
a,⇑
, Willie W.-H. Soon
b
, Nancy E. Quaranta
a
a
Comisio
´n de Investigaciones Cientı
´ficas de la Provincia de Buenos Aires – Grupo de Estudios Ambientales, Universidad Tecnolo
´gica Nacional, Colo
´n 332,
San Nicola
´s (2900), Bs.As., Argentina
b
Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138, USA
Received 11 February 2020; received in revised form 15 April 2020; accepted 20 April 2020
Available online 5 May 2020
Abstract
In paleoclimatology, the concept of latitudinal insolation gradients (LIGs), reckoned in various ways, has received increasing atten-
tion regarding glacial/inter-glacial climatic transitions and oscillations. In particular, the Holocene, which permits the reconstruction of
past climatic proxies with an increasingly finer spatial and temporal resolutions, has provided evidence that suggests that LIGs are a key
forcing on climate at different timescales. Nevertheless, LIGs’ own dynamics (chiefly their variations in relation to astronomical param-
eters and geographical zones) and even basic definitions, have not been properly investigated, especially during the last part of the present
geological epoch. The main reason is the lack of broadly accessible, theoretical insolation data that account for short-term orbital vari-
ations (i.e., for describing sub-Milankovic
´-orbital forcing during the Holocene). Based on our latest astronomical-orbital solutions, we
present an in-depth discussion on the calculation of LIGs and their variations all through the Holocene and 1 kyr into the future. Our
results show a much more complex variety and behaviour of LIGs than those that were shown previously. We report that during the
studied period, daily LIGs in summer, around the solstitial days (both hemispheres), are strongly modulated by obliquity only at
mid-latitude band, whereas at tropical and polar bands LIGs are modulated by ‘‘precession”. Summer half-year LIGs for the Northern
Hemisphere show a marked modulation in out-of-phase sense with obliquity, just for the mid-latitude and polar bands. Surprisingly, this
competing effect between ‘‘precession”and obliquity also produces the fact that the southern counterpart of these LIGs are more mod-
ulated by ‘‘precession”than obliquity. In cases involving inter-band latitudes or different intra-annual lapses, they need to be examined
separately and carefully and the results could be very different from traditional presumptions. Our novel results and study are based on
the precise estimation of the duration of the orbital interval considered in the definition of LIGs. Our new study also avoids the diffi-
culties of insolation calculations regarding the relationship between orbital longitudes and time.
Ó2020 COSPAR. Published by Elsevier Ltd. All rights reserved.
Keywords: Solar irradiance; Insolation gradients; Milankovic
´-orbital forcing; Sun-Earth relationship
1. Introduction
The differential solar energy that the Earth receives
between low and high latitudes drives the climate system
on planetary scales through both atmospheric and oceanic
circulation and differential heat storage and release mecha-
nisms. In climate studies, the insolation (incoming solar
radiation) at certain latitude /is primarily quantified as
daily irradiation or mean-daily irradiance over a full solar
day; this is the main source of energy that the climate sys-
tem receives daily at a fixed latitude. For the same solar
day, the contrast of insolation between two different lati-
tudes produces a latitudinal insolation gradient, LIG,
which activates a global-scale spatial temperature gradient
(see, e.g., Lindzen, 1994). In a simplified picture, the
https://doi.org/10.1016/j.asr.2020.04.030
0273-1177/Ó2020 COSPAR. Published by Elsevier Ltd. All rights reserved.
⇑
Corresponding author.
E-mail address: gcionco@frsn.utn.edu.ar (R.G. Cionco).
www.elsevier.com/locate/asr
Available online at www.sciencedirect.com
ScienceDirect
Advances in Space Research 66 (2020) 720–742
atmosphere can be divided into a system of circulatory
cells, particularly at the Equator, 30,60
and 90of lati-
tude (for both Northern and Southern Hemispheres, NH
and SH). These terrestrial latitudinal bands (i.e., the trop-
ical band, 0–30; mid-latitude band, 30–60; and the
polar band, 60–90), are also excited and modulated by
the seasonal variations and annual accumulation of solar
energy. The insolation received at any latitude is not con-
stant through time, instead it varies by intrinsic solar fluc-
tuations (i.e., caused by the solar magnetic cycles) and by
geometrical causes owing to variations of both Earth’s
orbit and Earth’s axial tilt. This last mechanism is the core
of the theory proposed by Milankovic
´concerned the pale-
oclimatic and geological history (Milankovic
´, 1941; Berger,
1978), which is a general astronomical theory about cli-
mate. This theory relies on the control that three dynamical
parameters: the eccentricity, e; the longitude of the perihe-
lion (reckoned from the equinox of the date, i.e., corrected
by precession and, if short-term periods are taken into
account, also by nutation), -; and the obliquity, , namely
the astro-climatic parameters, have on the secular March of
insolation at a particular latitude, /
1
(see Murray and
Dermott, 1999, as a standard astronomical reference on
celestial mechanics). The success of Milankovic
´’s theory
was due to the confirmation that the main long-term peri-
ods of the orbital and Earth’s orientation parameters
involved in the theory, i.e., 19 kyr and 23 kyr (due to
‘‘precession”), 41 kyr (related to obliquity oscillations)
and 100 kyr (eccentricity), are strongly imprinted in terres-
trial geologic and oceanic sedimentary archives (e.g., Hays
et al., 1976; Imbrie, 1982). Nevertheless, paleoclimatic data
also clearly show that these periodicities are not always
present with the same relative intensity, i.e., different
‘‘Milankovic
´periodicities”explain the main variance of
different data sets. For example, the long 100 kyr periodic-
ity is almost exclusively present after the late Pliocene and
early Pleistocene transitional era (i.e., after 0.8 Ma ago).
Before that epoch (and at least for about 2 Myr), the 41
kyr periodicity, that is related to obliquity, seems the dom-
inant modulator. Such discordance (i.e., from a linear
forcing-response paradigm) has created a fair debate not
only concerning the mechanism that drives, within the cli-
mate system, the long-term insolation variations, but also
related to which specific orbital insolation forcing drives
the climate at a certain time span (see e.g., Raymo and
Nisancioglu, 2003; Loutre et al., 2004; Liu et al., 2008).
Milutin Milankovic
´, following the views of his contempo-
raries, suggested that summer insolation at high latitudes
is the critical parameter affecting Northern Hemisphere
ice sheets (Milankovic
´, 1941, chapter XXI). From that pro-
posal, summer half-year or daily insolation for June (actu-
ally, at June mid-month; i.e., at the solstitial day) at high
latitudes (e.g., 65N) has been mainly used as an index of
the orbital forcing regarding paleoclimatic records (see
e.g. Imbrie, 1982). Nevertheless, it is well known that daily
insolation has a strong long-term modulation by preces-
sion
2
(see Hays et al., 1976, and also Eq. (28) herein
below), and as we have seen, the main precessional fre-
quencies are absent from some records of paleoclimatic
data.
Raymo and Nisancioglu (2003), in particular, proposed
using the difference between summer half-year insolation
between 25N and 70N as solar orbital forcing for the late
Pliocene/early Pleistocene interval. These authors showed
that obliquity has a long-term modulating effect on this dif-
ference. Nevertheless, this differential insolation, expressed
in W m
2
, i.e., in units of instantaneous or mean irradi-
ance, implies a temporal averaging of the differential irradi-
ation received in the half-year (in MKS system the natural
unit for irradiation is in J m
2
units). It is well known that
the duration of seasons or intra-annual lapses of insolation
depends, exclusively, on eccentricity and ‘‘precession”(see
Loutre et al., 2004, and Eq. (19) of our paper), hence it is
not obvious that the same kind of modulations of these
LIGs, related to obliquity, can be observed in shorter tem-
poral lapses which we will focus in this paper, because of
the competing effect between ‘‘precession”and obliquity.
In addition, as insolation depends on the observer’s lati-
tude, then it is important to address for which latitudinal
bands the dominance of differential insolation by obliquity
could be stronger.
It is worth highlighting that the effects of varying differ-
ential solar heating can also be more precisely studied for
the present epoch of Holocene. Nevertheless, specific works
dealing with LIGs accounting for the orbital dynamics at
these shorter timescales are rare. In particular, the Holo-
cene makes it possible to consider the very short-term vari-
ations (i.e., short-term Milankovic
´or often termed ‘‘sub-
Milankovic
´”forcing which ranges from interannual, deca-
dal to multi-millennial timescales) in orbital solutions
(Loutre et al., 1992; Cionco and Soon, 2017a). This is
important because the spatial and temporal resolutions in
the reconstruction of proxy data used to study climatic
variability have been comprehensively improved in this
modern epoch (Kirby et al., 2010; Shuman et al., 2018).
For example, in our short-term orbital solutions we are
able to account for the 18.6 yr lunar nodal cycle, originated
in the retrograding motion of the lunar orbit. For the few
rare exceptions, Davis and Brewer (2009), in a pioneering
work, discuss the long-term modulation of daily-solstitial
LIGs; i.e., the difference of mean-daily insolation at sol-
stices, just at the mid-latitude band. Using a pollen-based
reconstruction of temperature for the Holocene, they show
1
The semi-major axis aalso participates in insolation calculations but it
only has small short-term oscillations, it does not have secular variations
as the other elements, at least on a few million years time duration of
geological history.
2
The -angle is continuously modified by the progressive (very long-
periodic) effect of precession, among others. Then, and taking into account
the important spectral power that precession brings to long-term
insolation variations, it is usual in paleoclimatology to refer as dependent
on ‘‘precession”to all expressions dependent on -.
R.G. Cionco et al. / Advances in Space Research 66 (2020) 720–742 721
that these LIGs could have a more important role on tem-
perature gradients than previously thought. Nevertheless,
although they just consider the 60N-30N latitudinal band
(note the different sense ‘‘high-low”in LIG calculation here
in contrast to Raymo and Nisancioglu, 2003), they stated
in general that, summer LIGs depend on obliquity and
the winter ones depend on ‘‘precession”, with no other
specification or detailed discussion. Davis and Brewer
(2011), adopting the same latitudinal band, analysed these
solstitial LIGs at a short time period, from 1880 to the pre-
sent, considering other forcings and temperature gradient
simulations. They specifically show, for example, bidecadal
oscillations in reconstructions of latitudinal changes of
temperature in the Northern Hemisphere consistent with
the 18.6 yr lunar nodal cycle coming from the obliquity
part of mid-latitude LIG at mid-June. Hence, it seems that
at a much shorter timescale of years to decades, obliquity
also has an important role in our current geological epoch
of the ‘‘100 kyr world”.
We agree with all of these authors who argued, among
other issues, that the dynamics of LIGs have not been
widely investigated. Moreover, an open tool broadly avail-
able for calculating LIGs based on diverse temporal bases
(monthly, seasonal, annual, etc.) is needed. Taking into
account the importance of insolation gradients for the cli-
mate system, the lack of systematic research dealing with
this subject is surprising. In particular, one would expect
more studies that explore possible responses of climate
to various different LIG forcings especially since most
weather and climatic variations have roots in pressure gra-
dient forces. Loutre et al. (2004) marginally dealt with
annual LIGs, concentrating on the long-term (i.e., longer
than 1 kyr) annual irradiation variations. They show, for
example, that annual irradiation (and consequently its
associated LIGs) does not depend on ‘‘precession”(i.e.,
on the long-term variations of -), independently of the lat-
itudinal band considered; hence, they propose mean
annual irradiance as a plausible forcing strongly dependent
on obliquity. Cionco et al. (2018) re-calculated the same
daily LIG studied by Davis and Brewer (2011), showed
several peculiarities in their calculations and discussed
the detailed lunar and planetary short-term contributions
as well as the Total Solar Irradiance, TSI, variations that
accounted for the solar magnetic cycles over the last
2000 years. However, as far as we know, no other studies
on insolation gradients dynamics have been conducted for
the whole Holocene. Neither have we been able to find a
discussion of different kinds and combination of LIGs
for different latitudinal bands in the literature on climate
dynamics.
In this work we wish to discuss LIGs calculations from a
formal point of view, their time variations within the Holo-
cene and 1 kyr into the future, and to provide a standard
tool for LIGs and solar irradiation calculation taking into
account the latest astronomical solutions based on high-
precision planetary ephemerides. In Section 2, we re-
obtain and discuss the daily irradiation for a whole parallel
of latitudes, /, considering both total and mean-daily
irradiance.
In Section 3, we discuss the importance of orbital longi-
tudes (as a temporal reference) in the computation of daily
insolation along the year and how to use them from our
short-term orbital solution.
Section 4is devoted to the description of a summation
method to reckon intra-annual/annual lapses of irradiation
giving a special emphasis in the use of orbital longitudes.
Based on this formulation, we present in Section 5a formal
method to obtain LIGs for any intra-annual (e.g., daily
and seasonal) and also annual lapses. This exact definition
of LIGs is based on the calculation of the duration of the
orbital interval considered for each particular LIG. We
explain, for each type of LIGs, the most important short-
and long-term features. LIGs are fundamentally considered
as a forcing on climate, then it is necessary to acquire
knowledge regarding their modulations along the Holo-
cene; these are analysed in Section 6. The concluding
remarks are given in Section 7where we summarize our
findings in reference and with context to previous works
as well as potential applications. Finally, in the Appendix,
we provide a brief description of the computational tool we
offer and on how to obtain it along with the supplementary
files provided.
2. Re-deriving the insolation formulas for terrestrial parallels
of latitude
Insolation gradients are a measure of the differential
irradiation (difference in solar energy received during cer-
tain lapse of time) at the top of the atmosphere between
two latitudes. This description explains that the calculation
of the solar irradiation at different temporal lapses is the
core effort of LIGs calculation. The results of this Sec-
tion are of course not new, but they will help to better
understand key points in the procedure described in
Section 4.
Although the Sun irradiates all the elementary surfaces
at a particular parallel /, this holds only for the daytime,
with the instantaneous insolation Ið/;tÞ, at a particular
day, a function of the latitude and a discontinuous function
of time (it is considered zero at night):
3
Ið/;tÞ¼TSI 1au
r
2
ðsin /sin d
þcos /cos dcos HðtÞÞ;ð1Þ
TSI is evaluated as the solar irradiance received at a fidu-
cial distance (i.e., the astronomical unit, au, a fixed distance
to the Sun) on a surface perpendicularly oriented to the
Sun (reckoned, e.g., in W m
2
); dis the solar declination
and HðtÞis the Sun’s hour angle, which at a given date,
3
Defined as a step function, strictly zero at night, Ið/;tÞis integrable in
the Riemannian sense, with two removable discontinuities, at the moments
when the Sun rises and sets.
722 R.G. Cionco et al. / Advances in Space Research 66 (2020) 720–742
is solely a function of local time, because it evolves with the
diurnal movement of the Sun.
An elementary surface of the terrestrial globe centred at
latitude /has an area:
dr¼R2
cos /d/dLð2Þ
where Ris the Earth’s equatorial radius and Lis the ter-
restrial longitude reckoned/referenced from the originating
meridian (positive to the East). Considering the non-polar
zone
4
and following Milankovic
´(1941), we can relate the
Sun hour angle with the terrestrial longitude Lat which
the Sun is just crossing, but measured from the longitude
of an arbitrary observer, L0, at the latitudinal parallel /:
H¼L0L, which is negative to the East of the observer.
Then, the total irradiation, i.e., over the whole parallel,
for the full day of duration, s(in units of seconds, if TSI
is reckoned in W m
2
), is:
Q¼2TSI 1au
r
2R2
cos/d/
RL0
L1ðsin/sin dþcos /cosdcosðL0LÞÞdL Rs
0dt;ð3Þ
the problem has a clear symmetry around the noon of such
an observer (i.e., the daytime zone is symmetrical regarding
the mid-day; and because of this fact factor 2 appears in
Eq. (3)), then the integral over Lis better performed from
the noon of the observer (L¼L0), to L1, the farthest west-
ward longitude receiving daylight (dH¼dL), which
results in Eq. (3). On the other hand, elements other than
hour angle, are considered constant along the day (their
variations are very small regarding the change of H).
Hence, the total daily irradiation received by an infinitesi-
mal zonal strip of Earth’s surface of width, d/, is:
Q¼2sTSI 1au
r
2R2
cos/d/
ððL0L1Þsin/sin dþcos /cos dsinðL0L1ÞÞ:ð4Þ
The total irradiation (e.g., J m
2
) over a whole rotational
day per unit of surface area of zonal ring is:
Q/¼sTSI
p
1au
r
2
ððL0L1Þsin /sin d
þcos /cos dsinðL0L1ÞÞ;ð5Þ
L1is the longitude at which the instantaneous insolation
(Eq. (1)) ceases, that is:
sin/sindþcos /cos dcosðL0L1Þ¼0)cosðL0L1Þ
¼tan/tan d:ð6Þ
As we have centred the integral on the reference location of
the arbitrary observer’s noon time, the solar coordinates
(r;d, etc.) at mid-day need to be used in Eq. (5) and also
in Eq. (6), to determine the Earth’s daylight zone. Then,
as it is customary, we can replace L0L1by H0, the sunset
hour angle, which by symmetry, is equal in absolute value
with respect to the sunrise hour angle:
Q/¼sTSI
p
1au
r
2
ðH0sin /sin dþcos /cos dsin H0Þ;
ð7Þ
the expression for H0is obtained from Eq. (6). From this
geographical point of view, our derivations recover the
well-known formula of daily irradiation and clearly reveal
that the whole parallel of latitude /is being considered for
the calculation. Taking into account the diurnal disconti-
nuity of the irradiance and the rotational symmetry of
the problem, Eq. (7) is coincident with the daily irradiation
of any surface element for each particular observer at lati-
tude /.
Eq. (7) is usually completed with the calculation of the
mean-daily irradiance, which can be obtained just by aver-
aging Eq. (1) over Hangle and considering the other ele-
ments constant over a whole day (see also, Laskar et al.,
1993):
W/¼1
2pRp
pIð/;tÞdH¼TSI
p
1au
r
2
ðH0sin /sin dþcos /cos dsin H0Þ;ð8Þ
where, again, the symmetric integral is considered by tak-
ing the position of the Sun at noon time of the observer.
Another important result that we came up with by follow-
ing Eqs. (7) and (8) which in turn was first obtained by
Milankovic
´(1941):
W/¼Q/
s:ð9Þ
As a result, mean-daily insolation is independent of the
irregularities of Earth’s rotation through time. It corre-
sponds to the mean insolation of parallel of latitude /
for a specific day.
For the polar zones, the daily insolation is null for the
polar night. For the polar day, there is no day-night termi-
nator zone, hence the mean-daily irradiation, for example,
is obtained integrating Eq. (8) between 0 and 2p; then:
W¼TSI01au
r
21
2pR2p
0ðsin /sin dþcos /cos dcos HÞ
dH¼TSI01au
r
2sin /sin d:ð10Þ
In this case, the total irradiation, Q, is equally obtained
through the Eq. (9).
3. Using solar orbital longitudes as temporal reference along
the year
As Earth’s days go by, the Sun’s positional coordinates
and the astro-climatic parameters vary, although these
parameters change much more slowly. The modelled posi-
4
It is defined as latitudes satisfying j/j<90, i.e., where the Sun
rises and sets every day of the year.
R.G. Cionco et al. / Advances in Space Research 66 (2020) 720–742 723
tion of the Sun in its apparent geocentric orbit is controlled
by the true orbital longitude,k(see Cionco and Soon,
2017a, for details). For each day, kchanges about 1(at
least 0:7). The true longitude is important because it
determines the exact moment at which relevant seasonal
events as equinoxes or solstices occur. Unfortunately, there
is no fixed, unique relationship between calendar days and
k; therefore, the determination of when, for example, the
first equinox of the year occurs, or when, regarding the civil
calendar, the Sun acquires certain coordinates, is not a triv-
ial matter (see Laskar et al., 1993; Berger et al., 2010). The
arbitrary/inaccurate determination of the origin of orbital
longitudes produces the well known calendar problem: the
phase confusions for the insolation calculation in relation
to different origins of orbital longitudes (Joussaume and
Braconnot, 1997; Timm et al., 2008; Chen et al., 2011;
Bartlein and Shafer, 2019). Therefore, in consonance with
these authors, the best strategy to calculate periods of inso-
lation is to employ orbital longitudes, instead of calendar
days. Then, we can re-write Eqs. (7) and (8) making their
dependence on kexplicit, taking into account the well
known relationship of solar declination, H0(Eq. (6)) and
the Earth–Sun distance on k:
sin d¼sin ksin ;cos H0¼tan /tan d;
r¼að1e2Þ
1þecosðk-Þ;ð11Þ
therefore, we consider, for a particular day, the daily inso-
lation for the whole parallel of latitude /, parametrized by
a reference value of orbital longitudes:
Q/¼Q/ðkÞ;W/¼W/ðkÞ;ð12Þ
having in mind that formulas (7) and (8) were obtained
considering the constancy of angle dalong the day, and
therefore kas a fixed parameter within the day. Note that
the set of Eqs. (11) describes the functional relationship of
daily insolation with the astro-climatic parameters. It can
be clearly seen that obliquity variations just affect solar
declination, whereas precession, implicitly appears through
the Earth–Sun distance parameter, with a secular increase
in -angle. In this regard, one should keep in mind that
nutational modulation is also present at short timescale
(i.e., with periods less than 10 kyr), when short-term per-
turbations are considered in the orbital solutions (see
Cionco and Soon, 2017a).
In order to calculate the daily LIGs, we need to select
specific orbital longitudes within each year of the Holocene
and 1 kyr into the future. Here we follow the approach of
Cionco and Soon (2017a) where those authors obtained
and tabulated Earth’s astro-climatic parameters and corre-
sponding true orbital longitudes for each day of the whole
time interval considered (Cionco and Soon, 2017b). The
Earth’s true orbital longitude is tabulated at 12 h of each
day (i.e., a fixed lapse of 86400 s) in TDB (the dynamical
barycentric time) TT timescale (the corresponding rela-
tivistic terrestrial time). Hence, this approach is optimum
for describing the short-term orbital forcing, and we named
this accurate orbital solution STOF. In addition, STOF
solution permits directly the application of true orbital lon-
gitudes as an independent reference.
For a precise determination of the apparent position of
the Sun with reference to an observer on Earth, TT time-
scale must be related to UT1 (Universal Time), the pure
rotational timescale which only has irregularities in the
length of day (Morrison and Stephenson, 2004). Neverthe-
less, as the difference between UT1 and UTC, the universal
coordinate time (the hybrid quasi-uniform civil timescale),
is less than 0.7 s, our use of civil timescale will
adequately serve our purposes. It is related to TT by
UTC = TT - 32.184 s - Nls, where Nls is the number of leap
seconds introduced to keep UTC near UT1. Then, in our
model of a uniform day of s¼86400 seconds, the universal
time of solar culmination obtained from STOF solution is
11h59m27:82s. This corresponds to a meridian passage of
the Sun at a terrestrial longitude very near Greenwich
meridian, exactly at a longitude 080200:7. We call this
putative observer, ‘‘observer O”, that for practical pur-
poses, we consider at the origin of terrestrial latitudes. As
a consequence, using STOF solution we are able to get,
for each day, the true solar orbital longitude at the noon
of the reference observer O.
Nevertheless, temporal lapses and orbital arcs can be
formally related by the angular momentum integral
(Kepler’s second law), through the true anomaly angle, f:
2p
Tdt¼r
a
2df
ffiffiffiffiffiffiffiffiffiffiffiffiffi
1e2
pð13Þ
where Tis Earth’s orbital period, which is a function of
Earth’s semi-major axis that only deviates with small oscil-
lations around its mean value. The true anomaly is defined
as k¼fþ-(see Murray and Dermott, 1999, as a
reference on these astronomical definitions). Regarding
the present calendar, it is defined to follow the tropical year
as closely as possible (the lapse between two passages of the
Sun by the same equinox). Hence, the polar angle kdeter-
mines equinoxes and solstices at fixed months: the vernal
equinox (origin of orbital longitudes) corresponds to k=
0(March), June solstice is when k=90
(i.e., the Earth
is near its aphelion), September equinox is when k=
180, and December solstice when k= 270(i.e., when
the Earth is near its perihelion). Such calendar/krelation-
ship will be valid strictly for only a few thousand years
within the constraints of the incommensurabilities and
the slowly varying nature of astronomical cycles.
As a final note, Cionco and Soon (2017a)’s STOF solu-
tion is tabulated as a function of the Julian Day, then the
corresponding calendar day can be straightforwardly
obtained in relation to the present reference epoch.
4. Intra-annual and annual irradiation
Before discussing LIGs calculations, it is imperative to
take into consideration the estimate of the solar irradiation
724 R.G. Cionco et al. / Advances in Space Research 66 (2020) 720–742
for intervals longer than one day. If an integer number of
days, N, is under scrutiny, Eqs. (9) and (12) guarantee that
the irradiation of the interval can be obtained as the sum-
mation of daily irradiation/mean-daily irradiance for each
day of the period.
Milankovic
´(1941) already realised that when more than
one day is involved in the irradiation of a whole parallel,
the calculation is more subtle. Using intuitive arguments,
he based the calculation of intra-annual irradiation on
the integration of the mean-daily irradiance, considered
as a continuous function of the orbital longitudes along
the days. Berger et al. (2010) refined this procedure using
elliptic integrals, and they proved that this integral tech-
nique gives the same result as a summation of daily
irradiation.
In our approach, which is based on the use of true orbi-
tal longitudes, such intra-annual lapse (e.g., seasonal) is
specified by the first, k1, and the last, kNlongitudes of the
period, according to the procedure described in Section 3.
Then, the irradiation, S/, received during the correspond-
ing time interval defined by the first and the last orbital lon-
gitude of that lapse, is:
S/ðk1;kNÞ¼X
N
i¼1
Q/ðk0iÞ¼sX
N
i¼1
W/ðk0iÞð14Þ
where k0iis the solar orbital longitude at the noon of obser-
ver O for the day at which k1;kNand all the intermediate
values occur. Eq. (14) implicitly assumes N integer days
in the calculation, i.e., including the total daily irradiation
of days 1 and N.
Nevertheless, a strict condition k16k6kNcan be con-
sidered, for example, for seasonal calculation, which means
that one starts to reckon the irradiation just at the moment
at which k1is attained, and ends at the moment at which
k¼kN(see Berger et al., 2010). For these intervals of irra-
diation based on non-integer number of days, it is impera-
tive to consider the fraction of the received irradiance for
the first and last days. Taking into account that the first
event, k1, occurs at time t1, we must integrate Eq. (3)
between t1and s, hence the whole parallel’s irradiation
for the first day is:
Q/ðk01Þ¼ðst1ÞTSI
p
1au
r
2ðH0sin /sin d
þcos /cos dsin H0Þ¼ðst1ÞW/ðk01 Þ:ð15Þ
For the last day of the period, the temporal integral needs
to be evaluated between 0 h and tN, the time of the solar
event kN:
Q/ðk0NÞ¼tNTSI
p
1au
r
2ðH0sin /sin d
þcos /cos dsin H0Þ¼tNW/ðk0NÞ:ð16Þ
This calculation can be visualized with the help of Fig. 1
where the diurnal March of the insolation is depicted
together with the mean-daily irradiance (as a step
function). Then, using Eqs. (14)–(16), the intra-annual
irradiation received between two orbital longitudes k1
and kN, with the strict condition k16k6kN, can be writ-
ten as:
S/ðk1;kNÞ¼ðst1ÞW/ðk01 ÞþsX
N1
i¼2
W/ðk0iÞ
þtNW/ðk0NÞ:ð17Þ
Taking into account the fact that the true orbital longi-
tudes are known at the noon of observer O, the starting
and end times t1and tNcan be obtained using Eq. (13),
integrating from the tabulated value of orbital longitudes
(at noon, say 12 h TT), to the desired orbital longitude
(k1or kN):
ti12h ¼T
2pZfi
f0
ð1e2Þ3=2
ð1þecos fÞ2dfð18Þ
with fi¼ki-0i;f0¼k0i-0i;i¼1;N. All the astro-
climatic elements (such as -0i) and mid-day orbital longi-
tudes involved are obtained from STOF solution for the
corresponding day at which k1and kNoccur. The integrand
can be expanded at, for example, second order in e, given
by:
Zð1e2Þ3=2
ð1þecosðk-ÞÞ2df¼f2esin f
þ3
4e2sin 2fþOðe3Þ;ð19Þ
hence if the proper integral is negative, the k1or kNvalue
searched occurs before the local noon of observer O (see
Murray and Dermott, 1999, as a reference about this
kind of expansions in astronomy). An important result
from Eq. (19) is that durations of orbital intervals depend
only on eand -, then these variables would also depend on
long-term changes (where precession participates) as well
as on short-term variations (where nutation is present).
For example, the duration of the summer half-year, dS,
(from k¼0to k¼180) is, in days, up to first order in e:
dS¼365:2423 1
22
pesin -
;ð20Þ
where 365.2423 is the present value of the mean tropical
year (seven significant figures) in days.
In order to compare our methodology with that pro-
posed by Berger et al. (2010), we show the intra-annual
irradiation between k1¼0and kN¼70for observer O
at latitudinal parallels of 5N, 55N and 85N. Fig. 2 shows
a good agreement with Berger et al. (2010). For our calcu-
lation, we use a new code, LIG-LONG.for, that is, in turn,
built from Cionco and Soon (2017a)’s codes, and which
calculates daily and intra-annual irradiation using the
STOF solution. The code also calculates LIGs, as they will
be later defined in Section 5. When k1and kNorbital longi-
tudes are settled, the program finds the corresponding day,
for each year, at which k1and kNoccur, and sums the daily
irradiation between these values. The code evaluates
R.G. Cionco et al. / Advances in Space Research 66 (2020) 720–742 725
formulas (7), (8), (17), (18), etc., and considers the first and
last partly irradiated days.
With the same summation method, it is possible to
quantify the annual amount of insolation received in a full
year. In order to draw a comparison with Loutre et al.
(2004), we calculated the so-called annual-mean insolation
at latitude /;Y/, which is defined as Eq. (17), with the con-
dition 06k6360, divided by the duration of the period,
ds, where drepresents the fractional days involved (real
number), in this case, the length of the tropical year:
Fig. 1. Sketch of the instantaneous irradiance variation at the latitudinal parallel /, over two consecutive days (day 1 and day N) of length s. The Q/
values are the total irradiation for those days, which are characterized by the tabulated true orbital longitudes k01 and k0N, respectively. These irradiation
values are equal to the horizontally dashed areas. The corresponding mean-daily irradiances are W/ðk01Þand W/ðk0NÞ. The trand tsvariables are the Sun’s
rising and setting times for each day. Times t1and tNare the moments at which the Sun acquires the true longitudes k1and kNwithin those days. The total
irradiation between these times is equal to the sum of the obliquely dashed areas.
Fig. 2. Irradiation received between 06k670at parallels 5N, 55N and 85N for the whole Holocene. The solid dots mark the values obtained by
Berger et al. (2010) using elliptic integrals and a different set of astro-climatic parameters. The maximal difference with our results is about 2 MJ m
2
. The
observed decreasing tendencies of irradiation are because of the diminution of obliquity with time. The effect is more pronounced at high latitudes. The
value of TSI used is 1366 W m
2
.
726 R.G. Cionco et al. / Advances in Space Research 66 (2020) 720–742
Y/¼S/ð0;360Þ
ds:ð21Þ
Eq. (21) as a function of the mean-daily irradiance, has the
form:
Y/¼1
d1t1
s
W/ðk01ÞþX
N1
i¼2
W/ðk0iÞþtN
sW/ðk0NÞ
!
:
ð22Þ
Fig. 3 shows the outputs of Eq. (22), for 30S, from
10,000 BC to AD 1950. It is virtually identical to Fig. 6c
of Loutre et al. (2004), where the annual-mean irradiation
for 30S was depicted. In what follows we will define LIGs
formally, taking into account the temporal basis involved
in their calculations.
5. Latitudinal insolation gradients calculation
Unlike latitudinal temperature gradients which are
defined as the rate of change of surface temperature with
respect to latitudinal bands (Jain et al., 1999; Soon and
Legates, 2013), it is assumed that LIGs represent the vari-
ation along time of certain differential insolation quantity,
but averaged over a time span, or in the same way, the dif-
ference of two mean insolation values (i.e., expressed in W
m
2
).
There are important cautions necessary when calculat-
ing LIGs for arbitrary intra-annual temporal lapses, such
as the seasons. For example, if we calculate a summer
half-year LIG for the NH, from the sum of mean-daily
irradiance values during an integer and fixed number of
calendar days, between March 21 and September 21, e.g.,
for 184 days, it will be just an approximation that does
not contain the explicit astronomical information about
the orbital interval over time. We do not only refer to the
moments at which the equinoxes occur, but also to the
exact duration of the seasonal interval (see Eq. (20)). This
fact imposes an important constraint for a general defini-
tion of LIGs, because if an arbitrary lapse of time is consid-
ered, the ‘‘precessional”component will always be present
in LIGs definitions.
In addition, the sense of the differential insolation con-
sidered for LIGs estimations does not seem to be clear
nor uniform in the literature. For example Raymo and
Nisancioglu (2003) defined a summer LIG in the latitudinal
sense as ‘‘low–high”.
5
Defined in this way, this LIG is in
anti-phase relation with obliquity, i.e., obliquity decrease
produces an increase in daily LIGs (more solar energy
imposed at equatorial band). But it is in-phase, for exam-
ple, with benthic d18O records, which implies that the larger
daily LIGs are, the greater the amount of ice in polar
zones. For a detailed representation of obliquity variations
the reader is directed to the abovementioned works or
directly to Section 6of this paper.
Loutre et al. (2004) also use, in general, the sense low–
high but considering the daily LIG between 65Nand
30N, the sense is ‘‘high-low”, which is in-phase with obliq-
uity (see Fig. 6f of their paper). Davis and Brewer (2011),
Fig. 3. Mean-annual irradiance at 30S over the years AD 1950 to 10,000 BC (i.e., from the most recent time in the left to the most distant past to the
right). The annual averaging is performed over the tropical year, as calculated from LIG-LONG.for code. The temporal axis has been inverted to draw a
comparison with a small portion of Fig. 6c shown in Loutre et al. (2004). The value of TSI used for the sake of this comparison is 1363 W m
2
.
5
Note the inversion of scale in their Figs. 3 and 5, which can be
confusing; also the label Mar21-Sep21 70–25N is a typo, it should be
‘‘25–70”in order to correct for the sense of the latitudinal difference.
R.G. Cionco et al. / Advances in Space Research 66 (2020) 720–742 727
based on that work, used the same high-low sense for daily
LIGs. In our opinion, defining LIGs by means of the differ-
ence of the corresponding insolation quantity (daily, sea-
sonal, etc.) between ‘‘lower – higher”latitudes, motivates
a physics-based interpretation of LIGs. This is because,
in general, positive gradients indicate a net flux from high
concentrations to low concentration values. Making an
analogy with Fourier’s heat conduction law (which is pro-
portional to the negative variation of temperature), positive
lower-higher LIGs imply a net meridional heat flux from
lower to higher latitudes.
In what follows, in order to unify the generic expressions
for LIGs, we present an exact definition of LIGs that is
simply based on the detailed reckoning of the temporal
lapses involved.
5.1. Daily LIGs
LIGs, on a daily basis, will be defined as the difference of
daily irradiation (characterized by the orbital longitude k)
at two different latitudes (/1the lower and /2the higher
one), divided by the length of the day:
Ddð/1;/2;kÞ¼Q/1ðk0ÞQ/2ðk0Þ
s
¼W/1ðk0ÞW/2ðk0Þð23Þ
where Eq. (9) was used; k0is again the orbital longitude at
noon of observer O for the day at which koccurs. It is
assumed that the whole day is considered for the calcula-
tion. As already highlighted, this shows that daily LIGs
can be calculated as the difference of the mean-daily irradi-
ance at two latitudes. Fig. 4 shows the daily LIG between
30N and 60N at the June solstice (Ddð30N;60N;90)).
The long-term LIG signal increases with time, indicating
that more energy is progressively being received on a sum-
mer day at 30N, following the obliquity reduction over the
Holocene (see Section 6). At short timescales, the main
nutational periodicity of 18.6 yr (see Davis and Brewer,
2011; Cionco et al., 2018) is evident as illustrated in the
insert of Fig. 4.
5.2. Intra-annual and annual LIGs
In a similar manner, we define intra-annual/seasonal
LIGs as:
DSð/1;/2;k1;kNÞ¼S/1ðk01;k0NÞS/2ðk01 ;k0NÞ
dsð24Þ
where drepresents, again, the fractional number of days
involved, i.e., the complete duration of the orbital interval
considered and it is calculated as:
d¼1t1
sþX
N1
i¼2
iþtN
s;ð25Þ
where the summation, which is equal to N2, accounts
for the integer days between the first and last partially irra-
diated days.
Fig. 5 shows the intra-annual LIG between 5Nand
55N for 06k670(DSð5N;55N;0;70)). The lunar
nodal cycle (18:6 yr period) is also clearly shown at a rela-
tively shorter time record as an insert in Fig. 5. Note the
Fig. 4. Daily LIG between 30N and 60N at June solstice (k¼90) over the whole Holocene. The dominant 18:6 yr lunar nodal cycle, here associated
with obliquity, is clearly present as illustrated by the insert. The amplitude of these oscillations is 0.039 W m
2
; the y-axis of the insert ranges from
2.1 W m
2
to 1.85 W m
2
.
728 R.G. Cionco et al. / Advances in Space Research 66 (2020) 720–742
displacement of this LIG’s minimum to 4500 BC when
compared to the daily LIG of Fig. 4.
In order to be consistent with previous formalisation, we
define annual LIG as:
DYð/1;/2Þ¼S/1ð0;360ÞS/2ð0;360Þ
ds;ð26Þ
where dsin this case, is equal to the length of the tropical
year. Then using the mean annual insolation as it has been
defined in Eq. (21):
DYð/1;/2Þ¼Y/1Y/2;ð27Þ
which is equal to the mean-annual LIG used by Loutre
et al. (2004).InFig. 6, we present the annual LIG between
30S and 60S to compare it with Loutre et al. (2004)’s
Fig. 6d. The results are fully equivalent and self-
consistent (note the change of the sense of the time axis
for a direct comparison). The DYð30S;60SÞLIG is posi-
tive and growing at present times, consistent with the fact
that the diminution of the obliquity tends to increase
annual-mean energy at mid latitudes.
6. LIGs and their modulations throughout the Holocene
From the discussion offered along this work, it is clear
that ‘‘precession”and obliquity are well differentiated
astronomical parameters which strongly drive the varia-
tions of insolation at different timescales. The real motiva-
tion of this section is to present an empirical description of
these LIGs considered as potential index for the orbital
forcing through the whole Holocene, i.e., regarding their
possible modulating effects on climate proxies. In this
regard, the mid-Holocene is of especially interest, because
it has been used to strongly relate climatic variability and
insolation changes (see e.g., Kirby et al., 2010; Mancini
et al., 2005).
Our description of LIGs is based on the fact that,
although ‘‘precession”and obliquity are, in general, always
present in their formulations, it is possible to characterize
the long-term modulation of LIGs during the Holocene
following the main features of the precessional or the obliq-
uity curves (i.e., stationary points, convexity and inflection
points) as empirical criterion.
6.1. Daily LIGs
For the non-polar zone, up to first order in e, Eq. (8) can
be expressed as:
W//TSIð1þ2ecosðk-ÞÞðsin /F1ð/;;kÞ
þcos /F2ð/;;kÞÞ;ð28Þ
where F1and F2are functions of latitude, obliquity, and
solar orbital longitude. These functions do not depend on
eccentricity, therefore their expressions came directly from
Eq. (8). The Eq. (28) is very useful to gain insight about
daily insolation and LIGs variations. It allows us to differ-
entiate the obliquity dependent part (related to F1and F2)
from the ‘‘precessional”component. For example, at
equinoxes:
W//TSIð12ecos -Þcos /;
where the minus sign holds for September equinox. Hence
the ‘‘precessional”signal drives daily insolation at equi-
Fig. 5. Intra-annual LIG between 5N and 55N for 06k70, over the full Holocene interval. The insert shows, between the years 1900 and 2050, the
quasi-bidecadal oscillation coming from the obliquity, originated in the lunar nodal cycle. The amplitude of these oscillations is 0.028 W m
2
; the y-axis
of the insert ranges from 54.6 W m
2
to 54.85 W m
2
.
R.G. Cionco et al. / Advances in Space Research 66 (2020) 720–742 729
noxes and it is stronger at equatorial zone (see Cionco
et al., 2018). As a consequence, daily LIGs at equinoctial
days, are totally independent on obliquity. At this juncture,
it is essential to keep in mind that the ‘‘precession”term
always appears through a sine or cosine function of the
longitude of the perihelion in combination with the eccen-
tricity as will be also explained in the next subsection.
6.1.1. Summer days
For summer days, a broad picture to study the compet-
ing effects between obliquity and ‘‘precession”can be
drawn by analysing W/at the summer solstices (June for
Northern Hemisphere, NH, k¼90, or December for
Southern Hemisphere, SH, k¼270). This analysis is qual-
itatively valid for days with karound 40of the corre-
sponding solstices. For the solstitial days, insolation
depends on ‘‘precession”through the factor
ð12esin -Þ, where the negative sign holds for the SH.
In this model, the term esin -drives the Earth–Sun dis-
tance at summer solstices, and as we have seen, the dura-
tion of the summer half-year; hence to point out the
combined importance of eand -on climate in the frame-
work of the Milankovic
´theory, esin -parameter is usually
identified as climatic precession.
6
Note that climatic preces-
sion also has short-term variations, through high frequency
variations in eand nutational oscillations on -.
The specification of astro-climatic parameters permits
the overall estimation of their influence on daily insolation,
nevertheless, it is indispensable to directly evaluate each
particular LIG for each latitudinal band, in order to detect
the most influential astronomical element. For example,
daily insolation for summer time has climatic precession
modulations imposing more strongly near the equatorial
band, because if /!0, the contribution of F1disappears
and F2! 1. This behaviour has a clear impact on the
daily LIG involving the tropical band. Fig. 7 depicts the
daily LIG between 0Nand30
N. It evolution along the
Holocene and 1 kyr into the future is similar to the climatic
precession, with an inflection point around 3500 BC. For
the same day (Fig. 8), the daily LIG between 30Nand
60N, i.e., Ddð30N;60N;90), is similar to obliquity vari-
ations but in anti-phase sense, with the timing for the min-
imum of this LIG being coincident with the maximum of
obliquity around 7500 BC.
To complete the description of the classical latitudinal
bands related to circulation cells, for summer days, we
now include discussion on the polar zone, where Eq. (8)
is not valid. Fig. 9 shows the daily LIG on summer solstice
between 60N and 90N (i.e., the polar band) whose gra-
phic is similar to climatic precession curve. Indeed, the gra-
phic of the Ddð60N;90N;90) LIG shows (and it is
computationally ascertainable) the characteristic inflection
point related to climatic precession around 3500 BC. It is
quite similar to the daily LIG involving the tropical band
(Fig. 7)).
Fig. 10 shows altogether the three daily LIGs analysed
for the three latitudinal bands; note their relative magni-
tudes and the same negative sign for them. One important
result that can be drawn here is that although a summer
Fig. 6. Annual LIG between 30S and 60S. The time axis has been inverted to allow comparison with Fig. 6dofLoutre et al. (2004). Both results are fully
consistent with each others. The 18.6 yr lunar nodal cycle, through obliquity modulations, produces a clear bi-decadal oscillation in the annual LIGs. The
amplitude of these oscillations is 0.01 W m
2
; the y-axis of the insert ranges from 129.04 W m
2
to 129.11 W m
2
. The value of TSI used for the sake of
this comparison is 1363 W m
2
.
6
Note that the retrograding motion of the equinoxes only has climatic
effect on a non-circular orbit.
730 R.G. Cionco et al. / Advances in Space Research 66 (2020) 720–742
(June mid-month) daily LIG was used as representative of
orbital forcing controlled by obliquity (Davis and Brewer,
2009; Davis and Brewer, 2011), this only strictly holds, at
least for the Holocene, for the mid-latitudinal band. Simi-
lar LIGs but involving polar or tropical bands have long-
term modulations similar to climatic precession, even dur-
ing summer days.
Regarding the hemispheric similitude/difference in daily
summer LIGs, the SH mid-latitude LIG shows the same
kind of modulations with NH, i.e., in-phase with the NH
Fig. 7. Climatic precession variations throughout the Holocene (top panel) versus daily LIG between 0and 30N at June solstice (k¼90) (bottom
panel). This tropical LIG’s long-term signal is dominated by climatic precession, i.e., it follows climatic precession with a measurable inflection point
around 3.5 kyr.
Fig. 8. Earth’s obliquity variations throughout the Holocene (top panel) versus the daily LIG between 30N and 60N at June solstice (k¼90) (bottom
panel). The LIG signal is modulated in anti-phase sense by the long-term variation of obliquity. As in the obliquity graphic, there is no inflection point in
the curve of this LIG.
R.G. Cionco et al. / Advances in Space Research 66 (2020) 720–742 731
variations, which is an expected result if the LIG is mainly
driven or controlled by obliquity (changes fully controlled
by obliquity need to be in-phase with respect to the equa-
tor). For the other LIGs, the SH counterparts are in
anti-phase with the NH variations. This is consistent with
LIGs markedly controlled by climatic precession, because,
as explained at the beginning of this section, the climatic
precession factor is of opposing sign regarding the equator
(see Milankovic
´, 1941; Davis and Brewer, 2009, for
discussions on hemispheric variations of daily insolation).
This behaviour can be observed well in Fig. 11.InTable 1
we summarized the modulations of daily summer LIGs
Fig. 9. Daily LIG at June solstice (k¼90) between 60N and 90N (bottom panel). It follows climatic precession (top panel) as the daily LIG involving
tropical zone, with the characteristic inflection point around 3500 BC. Note, as in Fig. 7, the more independent behaviour around t.he present time.
Fig. 10. Daily LIGs at June solstice (k¼90) for the three indicated latitudinal bands in NH. LIGs involving high latitudes or the tropical band exhibit a
variation strongly modulated by climatic precession. The mid-latitude LIG closely follows the obliquity variations (see the text for explanation).
732 R.G. Cionco et al. / Advances in Space Research 66 (2020) 720–742
against the astro-climatic parameters, in addition with all
LIGs discussed throughout Section 6.
6.1.2. Daily LIGs on winter days
For the winter solstice (June for SH, k¼90, or Decem-
ber for NH, k¼270), the daily insolation signal is obvi-
ously weaker. This is so because F2is still positive and
the combination sin /F1, on the contrary, is negative for
both hemispheres. As a result, the obliquity dependent part
of daily irradiation is diminished regarding the summer
values, with a relative increasing role of the climatic preces-
sion component. The corresponding effect on winter LIGs
can be seen in Fig. 12, and is quite similar for all the winter
days. It is worth highlighting that the absolute value of the
winter LIGs is larger than that of their summer counter-
parts, because, at wintertime, insolation involving mid-
and higher latitudes is very small (i.e., only a few W m
2
).
The SH winter daily LIGs, because the climatic preces-
sion term has opposing sign for the two different hemi-
spheres, depend on climatic precession in anti-phase sense
with respect to the corresponding NH LIGs. See the sum-
mary for the modulations by astro-climatic parameters in
Table 1.
6.2. Intra-annual LIGs
On the intra-annual scale, for example seasonal, solar
irradiation is practically independent of climatic preces-
sion. This is a basic consequence of Kepler’s second law.
The solar flux, dQ, received at a specific time interval on
a surface facing the Sun, always depends on the inverse
squared law: dQ¼TSIð1au=rÞ2dt(see Eq. (1)). Because
the equinox is an almost fixed point along the year, we
can approximate Eq. (13) as dt/ðr=aÞ2dk, hence
dQ/TSIdkand the integration of intervals of irradiation
does not depend on the Earth–Sun distance. Then the sum-
mation of daily irradiation is practically not linked to the
relative positions of the equinox. Another consequence of
this fact is that theoretical approximations used to reckon
intra-annual irradiation (discussed in Section 4), which
are based on the integration of the mean-daily irradiance
as a continuous function of the orbital longitudes (Loutre
Table 1
Long-term modulations of LIGs discussed in Section 6in terms of
obliquity (O) and climatic precession (P) curves. In summer, seasonal
LIGs show a more subtle variation when compared to daily LIGs (related
to solstices): For the NH, the long-term modulation of LIGs for mid- and
polar bands follows the tendency of obliquity, but with visible changes
(delays) in the minima of these LIGs (hence, the Odlabel); for the SH,
these summer half-year LIGs have a clearer precessional influence that is
in-phase with climatic precession. The modulation of annual LIGs by
obliquity (Loutre et al., 2004), in an anti-phase relationship, is fully
confirmed for both hemispheres. The negative sign indicates a clear
(exact/close) anti-phase relationship with the corresponding parameter.
Daily (solstitial) Seasonal (half-year) Annual
Summer/Winter Summer/ Winter
60N-90N P/P -Od/P -O
30N-60N -O/P -Od/P -O
0-30N P/P P/ P -O
0-30S -P/-P P/-P -O
30S-60S -O/-P P/-P -O
60S-90S -P/ -P P/-P -O
Fig. 11. Daily LIGs at December solstice (k¼270) for the three indicated latitudinal bands in SH. LIGs involving high latitudes or tropical bands exhibit
a variation strongly modulated by climatic precession. These SH LIGs aree in anti-phase with the corresponding NH LIGs. As expected for a LIG driven
by obliquity, the mid-latitude LIG closely follows the obliquity variations that are in-phase with the corresponding NH LIG (see the text for explanation).
R.G. Cionco et al. / Advances in Space Research 66 (2020) 720–742 733
et al., 2004; Berger et al., 2010), are totally independent of
climatic precession.
Nevertheless, intra-annual LIGs behave differently com-
pared to intra-annual irradiation or differential irradiation
between two latitudes. They also depend on precession,
mainly, because the temporal basis (i.e., the intra-annual/
seasonal lapse of time) is itself a function of the climatic
precession, as was discussed in Section 5(see also Loutre
et al., 2004).
6.2.1. Summer half-year LIGs
This climatic precession-obliquity combination yields
the fact that in summer (half-year) season LIGs evolve
more similar to the climatic precession for the 0-30N
band, with a clear inflection point at 3500 BC, but with
the minimum of this LIG slightly shifted since about
8000 BC, i.e., near the Holocene’s obliquity maximum.
This LIG is presented in Fig. 13 in addition to the other
summer season LIGs corresponding to the other two latitu-
dinal bands under study. For mid- and polar latitudinal
bands of the NH the LIGs’ signals show convex curves sim-
ilar to obliquity variations, however the minima of those
LIGs are not coincidental with the obliquity’s maximum
as in daily summer LIGs (see Fig. 8), they appear delayed
with respect to the timing of the obliquity maximum.
Owing to the subtle climatic precession interaction with
obliquity, the minimum for the 30N-60N band LIG is
set in at around 3500 BC, while the minimum of the sum-
mer half-year LIG involving the polar band is placed at
around 5500 BC (Fig. 14)); that is, these LIGs also display
convex curves but with their minima shifted in time (see
also Fig. 16 for a graphic of the mid-latitude NH band
LIG).
Surprisingly, the SH summer half-year LIGs are
strongly marked by the climatic precession for all the latitu-
dinal bands (Fig. 15). Fig. 16 shows the 30S-60S mid-
latitudinal band LIG for the SH: it follows climatic preces-
sion, with the characteristic inflection point around 3500
BC, but showing a more independent behaviour after that
time. Possibly, the competing influence of obliquity after
that time produces a not so concave curve (i.e., from the
characteristic sinusoidal shape of the climatic precession
term). This behaviour confirms that obliquity is not fully
dominant in summer LIGs because when obliquity drives
insolation, the effect is the same for both hemispheres.
At this juncture, it is important to explain that the
marked differences between the same half-year LIG and
its hemispheric counterparts, have origins in the variability
of the temporal interval considered in the definition of each
LIGs. For example, regarding the 30-60half-year LIGs
(both hemispheres), they involve very different orbital
intervals dfor the Northern and the Southern counterparts,
as we can see in Fig. 17. Thus the summation of lapses of
insolation is not equal in each hemisphere. On the other
hand, dacts in the denominator in these LIGs formulations
(Eq. (24)), and it is strongly dependent on climatic preces-
sion (Eq. (20)). As a result, there is a marked sensitivity of
these intra-annual LIGs to the temporal lapses considered.
Fig. 12. Daily LIG variation between 30N and 60N at December solstice (k¼270) over the Holocene interval. The signal at this mid-latitude zone
follows the climatic precession. Obliquity signal, especially the 18.6 yr lunar nodal cycle, is strongly attenuated even at short-term variations as can be seen
in the expanded insert shown for 1900 and 2050. The y-axis of the insert ranges from 203.04 W m
2
to 202.92 W m
2
. The amplitude of these irregular
oscillations can reach 0.08 W m
2
.
734 R.G. Cionco et al. / Advances in Space Research 66 (2020) 720–742
6.2.2. Winter season
For winter half-year season (/d<0), the correspond-
ing LIGs curves follow (as daily winter LIGs) the variation
of climatic precession, with SH/NH LIGs in an anti-phase
relationship. See the summary compiled in Table 1.
6.2.3. Other intra-annual intervals
It is important to note that the above analyses and dis-
cussion of intra-annual LIGs are limited to seasonal half-
year LIGs. Because of the dependence of intra-annual
LIGs on the temporal lapse considered, all other intra-
Fig. 13. Summer half-year LIGs (NH, 06k6180) for the three indicated zonal bands in NH. For the tropical zone, the signal is similar to climatic
precession curve, whereas the other LIGs follows obliquity variations but out-of-phase, with their minima delayed several thousands of years relative to
the obliquity maximum (see the text for discussion).
Fig. 14. Earth’s obliquity variation throughout the Holocene interval (top panel) and seasonal LIG between 60N and 90N for the summer half-year
(bottom panel). The LIG signal shows a full convex curve in an out-of-phase sense regarding obliquity variations. Note that the LIG minimum for this
polar band has shifted to 5500 BC, unlike daily LIGs at mid-high latitudes, which have minima at 7500 BC, related to the obliquity maximum.
R.G. Cionco et al. / Advances in Space Research 66 (2020) 720–742 735
annual lapses must be carefully studied as well. For exam-
ple, the summer half-year DSð30N;60N;0;180ÞLIG in
the NH, varies as a full convex curve similar to obliquity
(Figs. 13 and 16); nevertheless, this LIG has a very differ-
ent modulation (similar to climatic precession) when the
astronomical summer season (i.e., the orbital interval
between the solstice of June and the equinox of September)
is considered (see Fig. 18). This is a clear example of the
effect produced by the change of the duration of the tem-
poral basis used in the definition of the LIG. The same
Fig. 15. Summer half-year LIGs for the Southern Hemisphere (1806k6360) for the three indicated zonal bands. Unlike the NH LIGs, these curves
show the characteristic variations of the climatic precession parameter.
Fig. 16. Summer half-year LIG for the SH mid-latitude band (bottom panel), and the same LIG for the NH (middle panel). It is clearly seen that SH mid-
latitude band evolves similarly climatic precession (i.e., it has a measurable inflection point), but showing a more independent behaviour after about 3500
BC. Nevertheless, the NH LIG follows a constant convexity curve (no inflection point could be detected) but out-of-phase, with its minimum delayed to
3500 BC, around the climatic precession inflection point.
736 R.G. Cionco et al. / Advances in Space Research 66 (2020) 720–742
kind of change occurs in the corresponding mid-latitude
SH LIG. When the astronomical austral summer is
considered, the corresponding LIG exhibits a complete
modulation similar to obliquity variations, unlike
DSð30S;60S;0;180ÞLIG which evolves similarly to cli-
matic precession.
Concerning the signs of the intra-annual LIGs, for tem-
poral intervals lesser than 15 days, intra-annual LIGs can
exhibit varying signs along the studied time span (i.e., alter-
nating between positive and negative values), whereas for
longer durations (e.g., seasons) the sign remains constant
for the whole studied time span.
Fig. 17. Duration in days of the summer half-year interval for both hemispheres. Near the year 4000 BC, both durations of these intervals coincide. These
varying durations have a remarkable effect on half-year LIGs values and their hemispheric differences, mainly because they define the denominator of these
LIGs (see Eq. (24)).
Fig. 18. Earth’s obliquity variation throughout the Holocene and the seasonal LIG between 30N and 60N for the astronomical summer
(906k6180). Unlikely summer half-year LIG, this LIG’s signal is similar to the climatic precession one, but in anti-phase sense. See Section 6. for
discussion.
R.G. Cionco et al. / Advances in Space Research 66 (2020) 720–742 737
6.3. Annual LIGs
Unlike seasonal LIGs, annual LIGs (i.e., 06k6360)
do not depend on climatic precession because the duration
of the tropical year is practically constant through time
(i.e., it has a small-amplitude short-term oscillation but
the secular variation can accumulate to about ten seconds
for the last two millennia). In particular, annual-mean irra-
diation has a change of phase around 43(N and S) as was
found earlier by Vimeux et al. (2001). This fact is shown in
Fig. 19. Mean-annual irradiance at 42S (bottom panel) and 45S (top panel) showing the transitional phase behaviour at around 43of latitude (both N
and S) (see text for discussion). The value of TSI used for this figure is 1363 W m
2
.
Fig. 20. Annual LIGs for the three indicated zonal bands in NH. All the curves are growing from 7500 BC onward when the obliquity has passed a
maximum.
738 R.G. Cionco et al. / Advances in Space Research 66 (2020) 720–742
Fig. 19. Nevertheless, all annual LIGs studied (including
the most important latitudinal bands), have the same pos-
itive convexity, i.e., growing function of time from about
7500 BC, i.e., in anti-phase relation to the obliquity varia-
tion (see Fig. 20). This annual LIGs dependence on obliq-
uity appears to be a robust result.
To conclude Section 6, we condense our analysis of
LIGs on obliquity and climatic precession, as a function
of the three fine timescales examined in Table 1. In this
arrangement, we mark with O or P the dominant func-
tional form in the long-term modulation of the correspond-
ing LIGs.
7. Concluding remarks
This paper describes a method for calculating LIGs
throughout the Holocene and 1 kyr into the future, for
any location on the Earth. Differential insolation between
two latitudinal bands, characterized as LIGs, constituted
one of the most conspicuous insolation forcings proposed;
however, LIGs’ basic definitions and dynamics throughout
the full time domain of Holocene have not been widely
investigated. Our work attempts to fill in these conceptual
gaps, and provides a practical tool which is based on the
precise calculation of the orbital interval considered in
LIGs definition. We provide a complete and fast Fortran
program valid for both paleoclimates studies and present-
day calculations of theoretical irradiation intervals and
LIGs, ready for broad applications in the field of climate
dynamics and environmental modelling.
Regarding the sense of meridional gradient and trans-
port scenarios associated with the studied LIGs, our results
yield several conclusions, summarized as follows:
>Annual LIGs values are always positive, indicating a
net pole-ward flux of solar energy (see Fig. 21). They are
controlled (for any latitudinal band and for both hemi-
spheres) strongly by obliquity and in anti-phase relation-
ship. In this sense, our work both confirms and extends
the previous findings in Loutre et al. (2004).
>For summer time (i.e., the season adopted as the most
widely accepted orbital forcing in terms of a net balance of
ice-mass according to the Milankovic
´astronomical theory
of climate), the 0-30(tropical) band produces daily and
seasonal LIGs which are always negative, suggesting a
net equatorial flux of solar energy. For the other bands,
30-60(mid-latitude zone) and 60-90(polar), daily LIGs
are also negative, but seasonal half-year LIGs are positive,
resulting, in this case, in a net pole-ward flux of energy.
Hence, during summer, the tropical band always produces
negative LIGs on daily and seasonal bases, for the whole
Holocene; the other bands produce positive LIGs at
seasonal-annual scale, but not on daily bases. This point
is summarized in Fig. 21. An important consequence of
these differences in the meridional flux associated with
these LIGs is that the underlying physical processes in
the climate system, activated by these insolation changes,
need to be different. For example, positive LIGs are consis-
tent with a model of moisture transport and consequently
feeding of ice-caps (Raymo and Nisancioglu, 2003); never-
theless negative LIGs could be physically interpreted in a
model as heat removal mechanism from higher latitudes.
The long-term modulations of the studied LIGs, are
summarized in Table (1). Regarding summertime, we espe-
cially conclude that:
>Over the studied time span, daily LIGs at mid-
latitudes closely follows obliquity variations in anti-phase
sense, being more similar to climatic precession in the
another bands. Taking into account that daily LIGs in
the NH and in the SH are in-phase for mid-latitudes, these
LIGs seems to be driven or controlled by obliquity (see
Section 6). For the other studied bands, the SH counter-
parts are in anti-phase, which is an expected result if the
LIGs are really driven by climatic precession.
>The seasonal half-year LIGs show more complex vari-
ations regarding daily LIGs. In the NH, for mid-latitude
Fig. 21. Meridional sense of the fluxes associated with annual, daily (solstitial) and seasonal (half-year) LIGs analysed in Section 6for the indicated zonal
bands in summer time. Poleward fluxes (positives) are indicated with up (red) arrows and equatorward fluxes (negatives) are shown with down (blue)
arrows. In a model of e.g., moisture transport (Raymo and Nisancioglu, 2003), positive gradients are consistent with a feeding of ice caps or ice mass
accumulation. See Section 7for explanation. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of
this article.)
R.G. Cionco et al. / Advances in Space Research 66 (2020) 720–742 739
and polar bands, they evolve similarly to obliquity varia-
tions but in out-of-phase sense; whereas the same seasonal
LIGs for the SH, show similar variations to climatic pre-
cession. This result challenges the earlier claims by
Raymo and Nisancioglu (2003) and Davis and Brewer
(2011) that summer LIGs, in general, are dominated or
controlled by obliquity. Indeed, Fig. 22 depicts the summer
LIG between 25N-75N during the Holocene (already
studied by Raymo and Nisancioglu, 2003, for the late
Pliocene), where a clear signal owing to climatic precession
(anti-phase) can be observed, with the characteristic inflec-
tion point around 3500 BC, although a convexity more
related to obliquity can be seen after that time.
We would like to stress the effect produced by the vari-
ability of the temporal interval considered in the definition
of each intra-annual/seasonal LIG. This even produces
that the hemispheric counterparts of the same half-year
LIGs can have a very different variation along the Holo-
cene (see Table 1).
These empirical facts clearly show that the broad defini-
tion of LIGs as the difference between ‘‘low”and ‘‘high”
latitudes is not useful for the Holocene, and that a more
precise identification of the latitudinal bands and/or the
orbital lapses considered is mandatory. This also holds
for any intra-annual lapses considered other than half-
year season. For example, in Fig. 5, in spite of the use of
near equatorial (5N) and mid-latitudes (55N), the corre-
sponding LIG (for about the NH spring) is modulated fol-
lowing obliquity variations (unlike NH tropical band in
summer half-year, which follows an evolution similar to cli-
matic precession). This illustration means that any different
combinations of inter-band latitudes and intra-annual peri-
ods need to be carefully examined before one can propose
these metrics as potential climate forcings or drivers.
Our practical tool will enable researchers to compare the
time records of climate proxies with several insolation
quantities and including the relatively less studied SH’s
LIGs for the whole Holocene (see e.g., Vimeux et al.,
2001; Whitlock et al., 2007; Wagner et al., 2007; Tonello
et al., 2009; Bosmans et al., 2015; Kender et al., 2018). In
addition, our approach avoids directly the astrometric dif-
ficulties such as the well-known calendar problem
(Joussaume and Braconnot, 1997; Timm et al., 2008;
Chen et al., 2011; Bartlein and Shafer, 2019) and the need
to calculate mean orbital longitudes in order to relate orbi-
tal longitudes to time (Laskar et al., 1993; Berger et al.,
2010), for the whole Holocene and 1 kyr into the future.
In addition, our program permits users to calculate very
accurate LIGs and intra-annual and daily lapses of insola-
tion at the present times, hence it has a potential for a very
wide-range of applications that include, for example, the
Angstro
¨m-Prescott ground radiation models (Bueche and
Vetter, 2015; Despotovic et al., 2015; Wang et al., 2018).
Another compelling possibility, as suggested by the dis-
cussion about the displacement of summer LIGs’ minima
(e.g., Fig. 14 and Fig. 16), would be a systematic evaluation
and study of the role of these LIGs in explaining the timing
for the peak Holocene warmth intervals for high and mid-
dle latitudes and even tropical/equatorial regions (see, e.g.,
McKay et al., 2018; Sachs et al., 2018; Tarasov et al., 2018;
Thienemann et al., 2019; Wu et al., 2018; Zhang and Feng,
2018; Gao et al., 2019; Regne
´ll et al., 2019) that cannot
simply be explained by the direct in situ insolation forcing
factor alone. We hope the ideas expounded in this paper
Fig. 22. Earth’s climatic precession variation throughout the Holocene and seasonal LIG between 25N and 75N for the summer half-year. The signal is
similar to climatic precession in anti-phase sense. See Section 7. for discussion.
740 R.G. Cionco et al. / Advances in Space Research 66 (2020) 720–742
can contribute to broaden the application of LIGs so as to
gain insights in the evaluation of their intrinsic connections
to climate.
Regarding the magnitude of the LIGs, it is important to
note that the orbital forcing produces changes of several W
m
2
along the Holocene and these variations (which are
very reliable because astronomical models are very accu-
rate at these timescales) should be considered as mean fluc-
tuations over which the TSI variations (rooted in intrinsic
magnetic activity of the Sun) also need to be considered.
For details concerning this issue (especialy the relative
importance of TSI variations against orbital changes in dif-
ferent seasons), the reader is directed to Davis and Brewer
(2011), Cionco et al. (2018) and to the supplementary mate-
rial published in PANGAEA, which explains how our
LIG-LONG.for can be used to experiment with any recon-
struction of TSI or to specify any TSI value for the present
time.
Acknowledgements
The authors acknowledge the support of the grant PID-
4362 from Universidad Tecnolo
´gica Nacional of Argen-
tina. WWHS’s works are also indirectly supported by
SAO grant with proposal ID: 000000000004297- V102.
RGC further thanks Prof. Luciana Andrı
´n of UTN, for
her support on English editing. The authors are grateful
to anonymous reviewers for key suggestions which
improved the presentation of their results.
Appendix A
The Fortran code (LIG-LONG.for) is available in
PANGAEA data publisher (https://doi.pangaea.de/10.
1594/PANGAEA.903521) with detailed explanatory notes.
This program permits the calculation of insolation quanti-
ties and LIGs for any selected latitudes over the Holocene
and 1 kyr into the future, given the corresponding orbital
longitudes k1and kN(for daily insolation k1=kN). The
core effort of the code is the finding of the solar orbital lon-
gitude at the noon of observer O for the day at which k1;kN
and all the intermediate values occur. It also allows users to
specify non-constant TSI values with time. Then, this soft-
ware is a new tool for solar/orbital forcing simulations.
The STOF solution is tabulated as a function of the
Julian Day, then the corresponding calendar day can be
straightforwardly obtained and related to true orbital lon-
gitudes using the software provided in Cionco and Soon
(2017b).
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