The Milankovitch astronomical theory of paleoclimates: A modern review

V~St~ZS in As~ronon~jj 1960, Voi.24, pp. I03-122. Pergamon Press Ltd. Printed in Great Britain. Editors: A. Beer, K. Pounds and P. Beer

THE MILANKOVITCH ASTRONOMICAL THEORY OF PALEOCLIMATES: A MODERN REVIEW Andr~ ]~.rger Institute of As~onomy and Geophysics, Catholic University of Louvain, Bel~um

I.

INTRODUCTION

When c l i m a t i c v a r i a t i o n s clearly

state

does f l u c t u a t e

and v a r i a b i l i t y

have t o be e x p l a i n e d , i t i s f u n d a m e n t a l t o f i r s t

t h e s c a l e one would l i k e t o c o n s i d e r , b o t h i n time and s p a c e . significantly

a way a s t o make one d e c a d e , one c e n t u r y o r one m i l l e n n i u m d i f f e r e n t ( B e r g e r , 1979a).

It

Indeed, climate

from one y e a r t o a n o t h e r , b u t a l s o i t g r a d u a l l y c h a n g e s i n such from t h e one b e f o r e

i s i n f a c t b e c a u s e our memories tend t o b e t o o s h o r t to r e c a l l

past

y e a r s t h a t we a r e a l a r m e d when a n u n u s u a l l y s e v e r e w i n t e r (1976-1977 i n t h e U . S . A . ) o r p r o l o n g e d d r o u g h t o c c u r s (1976 i n

W e s t e r n Europe and S a h e l ) .

Even i f t h e l o n g e r - t e r m r e c o r d s a r e s t i l l

being laboriously reconstructed,

t h e r e i s much

geo-ecol0glcal evidence that climate has fluctuated in the more distant past. particularly true for the last Ice Age.

This is

Figure I shows indeed that this ice age displays

quite a lot of climatic variations, but even more particularly striking is the fact that these natural changes are essentially characterized by their respective amplitude : the mean rate of temperature variation over very long extended periods is of t h e order of 10aC/10,000 years, while over the historical period, it reaches around 1.5°C/100 years; since 1940, the mean rate of cooling, which is still in progress, is 0.0TC/year.

These y e a r l y f l u c t u a t i o n s

and l o n g e r - t e r m c h a n g e s have b e e n t h e r e s u l t

of natural processes

a t work o n t h e complex system, which d e t e r m i n e s the E a r t h ' s c l i m a t e : a t m o s p h e r e , o c e a n s , c r y o s p h e r e , l i t h o s p h e r e and b i o s p h e r e . h a v e become a n o t h e r s i g n i f i c a n t

It

i s o n l y s i n c e a few y e a r s t h a t mankind a p p e a r s to

f a c t o r i n t h e c l i m a t i c b a l a n c e ( B e r g e r , 1980).

p r e s e n t r e v i e w s we w i l l o n l y f o c u s on c l i m a t i c v a r i a t i o n s

with characteristic

I0,000 to 100,000 y e a r s which m a i n l y r e p r e s e n t g l a c i a l and i n t e r g l a c i a l the quaternary p e r i o d .

From a t y p i c a l

J.zv.*. 24/2--A

periods of

oscillations

during

s p e c t r u m o f c l i m a t i c changes ( F i g . 2) e x t e n d i n 8 from

p e r i o d s c o m p a r a b l e t o t h e age o f t h e E a r t h t o about I h r , seem t o b e r e l a t e d

In t h i s

these Quaternary ice-volume c y c l e s

t o o n l y one t h e o r y , t h e M i l a n k o v i t c h a s t r o n o m i c a l t h e o r y o f p a l e o c l i m a t e s .

! 03

A. Berger

104

Northern Hemisphere annual mean temperature

Summer temperature in Europe

~l.O*C

-0,8 0

11170

,~o •

lSSO

08

~

(•)

Eem

(~) Holocene

.

®

,.o-® 1910~

"® t

(~

Oryos

(~

Climatic Optimum

(~) Little climatic optimum ( ~ Little Ice Age

A-D.

A.O.

(~) T~e

--i5"

1940's

C

,-. iO.C |

n-

12

IO

O

IO

14

O

•

190 -

s-

=?e-

=4.0

-Z.O

18O

- ~ 245

31Ft. 475-

"

IO •

i,

~1| 7 -

•

607"

=o. 14o

January temperature in Chile

Fig. I.

2.

Summer temperature Northern Atlantic

7=*.

~04 51'8

"~ ~

La,o ='

! 747

Ice volume

Climatic changes during the Quaternary ice age (adapted from U.S. Federal Council for Science and Technology, 1974; paleoclimate d e t a i l e d information a v a i l a b l e in B e r b e r , 1979a).

ASTRONOMICAL ELEMENTS OF THE EARTHtS ORBIT

The astronomical theory of paleoclimates relates the climatic ,variations to those in the solar energy which would be available at the Earth's surface for a completely transparent atmosphere.

For any latitude on the Earth, this insolation (Berger,

1975) is a single-valued

function of the solar constant SO, of the seml-major axis u of the ecliptic, of its eccentricity e, of its obliquity ¢ and of the longitude ~ of the perihelion measured from the moving vernal equinox (Fig. 3).

The secular perturbations used in ordinary astronomical practice to compute these orbital elements of the Earth (expansion in powers of the time) are not appropriate for our problem, because they are adequate only for some centuries centered on the present.

If information

over longer intervals of time is desired, an analytical solution expressed in trigonometrical form is required, a solution the accuracy of which depends essentially upon the accuracy and the number of terms kept in the perturbation function of the Lagrange and Laplace equations.

Analysis of the differential equations of the planetary motion shows that those trigonometrical solutions progress in powers of the following small parameters:

(I) the ratio

of the masses of the planets to that of the sun, and (2) their eccentricities and inclinations to the reference plane.

There are thus two kinds of approximation.

The first

The Milankovitch Astronomical Theory of Paleoclimates: A Modern Review

i n v o l v e s the d i s t u r b i n g m a s s e s and t h e second i n v o l v e s the p l a n e t a r y e s and i s . solutions are dependent to the first

105

I n t h i s way,

o r t o t h e s e c o n d o r d e r on t h e m a s s e s , and t o the f i r s t

o r to t h e t h i r d d e g r e e on the p l a n e t a r y e s and i s i f ,

respectively,

terms o f o r d e r e q u a l to

and h i g h e r t h a n two o r t h r e e i n masses and terms o f d e g r e e e q u a l t o and h i g h e r t h a n t h r e e o r f i v e i n e s and i s a r e n e g l e c t e d ( e v e n - d e g r e e terms do n o t e x i s t , the p e r t u r b a t i o n

because the expansion of

f u n c t i o n o f t h e p l a n e t a r y s y s t e m does n o t i n c l u d e odd powers o f the e s and

is). In f a c t ,

t h e s e s o l u t i o n s a r e o b t a i n e d from a s s u m p t i o n s made t o s o l v e the c e l e s t i a l

fundamental equations i n e c c e n t r i c i t y fixed vernal point, the a s c e n d i n g n o d e .

inclination

l o n g i t u d e ~ o f t h e p e r i h e l i o n measured from the

i o f t h e o r b i t on the r e f e r e n c e p l a n e , and l o n g i t u d e ~ o f

As m a t h e m a t i c a l t e c h n i q u e s f o r s o l v i n g d i f f e r e n t i a l

l e a d i n g t o s o l u t i o n s w i t h a l a r g e r and l a r g e r number o f t e r m s .

of observations is also improving, allowing a better the initial

e q u a t i o n s a l l o w more

n u m e r i c a l c o m p u t a t i o n s , t h e a s s u m p t i o n s n e e d to be l e s s and l e s s

and more s o p h i s t i c a t e d restrictive,

e,

conditions,

As t h e q u a l i t y

d e t e r m i n a t i o n o f t h e c o n s t a n t s and o f

t h e s e t e c h n i q u e s o u g h t t o p r o v i d e more and more a c c u r a t e s o l u t i o n s .

/

/Plone,ory Evolution

Orbital vorio~ions

® Io3

10=

Tectonic j

/

Annual

l j

104

~

/

R~=0on

Char.Time o • r Chot. Time Diurnol / Scale ~mp-sea Arctic SeaRotation J Antarctic ~ l i ~ ~ IThewn Relaxation ~ clrcu,l~lons Solar octivityJmlxed layer J

,o

Imonth

Icloy

J 10 8

I0 e

I0 7

I0 6

105

I0 4

tO3

I0 =

I0 J

I

I0-I

I I0"2

10.3

Time in years

Fig. 2.

mechanics

Tentative spectrum of climatic variations. Estimate of relative variance of climate over all periods of the variation. A background level of variability, deriving from internal stochastic mechanisms and corresponding to a low degree of predictability, appears to increase in amplitude towards the l o n g e r time s c a l e s and t o b e o v e r - t o p p e d by b a n d - l l u ~ t e d v a r i a b i l i t y due t o e x t e r n a l f o r c i n g p r o c e s s e s and c o r r e s p o n d i n g to a high degree o f p r e d i c t a b i l i t y ( a d a p t e d from M i t c h e l l , 1976).

106

A. Berger

S

Fig. 3.

Elements o f the E a r t h ' s o r b i t . The o r b i t o f the Earth, E, around the Sun, S, i s r e p r e s e n t e d by the e l l i p s e PTE4, P being the p e r i h e l i o n and A the a p h e l i o n . I t s e c c e n t r i c i t y , e, is given by (~2.~2)i/2/~, a being the semi-major axis and b the semi-m/nor axis. WW and SS are respectively the winter and the st---~r solstice, 7 is the vernal equinox; WW, SS and 7 are located where they are today. SQ is perpendicular to the ecliptic end the obliquity, e, is the inclination of the equator upon the ecliptic --i.e. ¢ is equal to the angle between the Earth's axis of rotation SN and SQ. ~ is the longitude of the perihelion relative to the moving vernal equincx, and is equal to H+¥. The annual general precession in longitude, ¥, describes the absolute motion of Y along the Earth's orbit relative to the fixed stars. ~, the lon~tude of the perihelion, is measured from the reference vernal equinox of 1950 A.D. and describes the absolute motion of the perihelion relative to the fixed stars. For any ntm~erical value of ~, 180" is subtracted for a practical purpose: observations are made from the Earth, and the Sun is considered as revolving around the Earth.

A f i r s t - o r d e r s o l u t i o n f o r the e i g h t p r i n c i p a l p l a n e t s was a l r e a d y g i v en by Stockwell* more than I00 years ago.

This Lagrange type o f s o l u t i o n i s obtained by the very d r a s t i c process

o f d i s r e g a r d i n g the p e r i o d i c terms i n the development o f the d i s t u r b i n g f u n c t i o n and l i m i t i n g this disturbing function to the quadratic terms in the es and is (~2).

The system of

differential equations for the elements of the principal planets then breaks up into two separate systems, one for ecos(~), esin(~) and one for sin(i)cos(~), sin(i)sin(~).

The

expressions are linear homogeneous differential equations of the first order with constant coefficients,

a determinant equation furnishing the periods of the individual terms.

In 1950, the improvement brought about by Brouwer and van Woerkom was attained through the use of more accurate planetary masses (Newcomb)

and the evaluation of a second-order effect

produced by the great inequality in the motions of Jupiter and Saturn.

Indeed, Hill showed

that the terms in the disturbing function for Jupiter and Saturn that arise from the great inequality 2 Ij - 5 IS have a considerable influence on the secular variations of the eccentricities and perihelion, an influence which is at least as important as the one coming from the long-period terms of degree three originating directly from the secular part of R. It is thus the reason why Brouwer-van Woerkom and Sharaf-Budnikova have considered the Lagrange solution with, for Jupiter-Saturn,

the Hill-Leverrier disturbing function where

terms up to the degree six in es and is are taken into account.

These short-period terms

generated long periods, of which Brouwer kept only the two most important ones.

*A full list of references is given in Berger (1976, 1977a).

This

The Milankovitch Astronomical Theory of Paleoclimates: A Modern Review

]07

inclusion caused an appreciable change in one of the terms in the eccentricity-perihelion (e,~) solution: Period

Coefficient

S tockwell

348,700 year

0.01634

Brouwer-van Woerkom

30 ], 700 year

O. 01834

Although this new representation of the long-tern variations of the Earth's orbital elements, when compared to Stockwell, shows differences up to 30% in various terms, the character of the variation remains the same.

As t h e e s and ~s a r e s m a l l , t h e n e g l e c t e d s e c u l a r terms o f the p e r t u r b i n g f u n c t i o n whose d e g r e e s w i t h r e s p e c t t o t h e e s and i s a r e h i g h e r t h a n t h e s e c o n d a r e a l s o s m a l l as compared with those not neglected.

At the same t i m e , however, i t

additional

term in a differential

especially

if

is possible

e q u a t i o n may have a n o t a b l e e f f e c t

the time be s u f f i c i e n t l y

extended.

that a very small in the integral,

A l t h o u g h L e v e r r i e r and Harzen were a l r e a d y

t a k i n g i n t o a c c o u n t t h e s e terms o f h i g h e r o r d e r i n t h e s e c u l a r p e r t o f the d i s t u r b i n g i t was A n o l i k e# aZ. who, f o r the f i r s t

time i n 1969, s y s t e m a t i c a l l y

considered all

terms o f t h e d i s t u r b i n g f u n c t i o n up to the f o u r t h d e g r e e i n e s and i s

for all

function,

long-period

the e i g h t

planets.

Thus, as important modifications of these developments may arise from commensurabilities among the periods of the principal planets, the effect of near-resonances must be included, and it was in ]974 that Bretagnon made a decisive improvement in the evaluation of e, ~, i, ~.

The main characteristics of that solution are as follows.

The elements of the

Earth's orbit are referred to the mean ecliptic and the mean equinox of ]850.0.

The initial

elements at this epoch of reference are the most accurate available, and the most recent determination of the planetary masses (Kovalevsky, 1971) is used.

Constants of integration

have been care~ully determined in such a way that the solution coincides with the elements at the time origin ;850.0 (a calculation which is not found in a similar work by Anolik e~ ul., ]969).

About the disturbing function, in addition to all the long-period

terms up to the fourth order in planetary es and is, BretaEnon has introduced some terms coming from the short-perlod part of the disturbing function.

About the influence of these

short-period terms to the second order as to the masses, terms limited to the third degree in es and is, which lead to a modification of the lonE-period frequencies by more than ]0-3 "/year (around 50 for the only Jupiter-Saturn near-resonance), have been kept for all the planets.

Among the long periods generated through these terms, only those whose

amplitude is greater than lO -~ for the inner planets and ]0-6 for the major ones have been considered.

Among the short-period terms used, in particular those of order three of the

great inequality in the motion of Jupiter and Saturn are introduced.

In addition to e, ~, i and ~, the paleoclimatic problem requires also the obliquity e and the annual general precession in longitude ~.

For e and $, Poisson equations have been solved

through the Sharaf-Budnikova (;967) method, where the series expansions include terms up to the second degree in eccentricity.

This finally allowed Berger (1976) to determine the

long-tern variations of the elements of the Earth's orbit as required in paleoclimatology: this solution for

(e,~,i,~) includes terms to the second order of the disturbing -~sses, and

to the third degree with respect to the planetary es and is; for c and %, Sharaf's analytical expansions were used, but their numerical determination takes into account all the Bretagnon terms.

108

A. Berger

Critical analysis of theories of the long-term variations of the elements of the Earth's orbit (Table I) and their numerical comparison leads to the following conclusions (BerBer, 1977a) regarding the influence of different terms on the accurac~ of the expansions used: (I) (2)

further improvement in planetary masses will not have significant influence~ for the (e,~) system, terms depending upon the second order as to the disturbing masses are more important than ones coming from the third degree with respect to the planetary ~s and gs;

(3)

for the (i,~) system, the latter terms have highly significant influence, whereas additional terms in masses are negligible.

(4)

The same conclusion can be drawn for (c,~);

the mest up-to-date solution (BerBer, 1978a) is close to the ideal one and probably can provide valuable information up to 5 million years.

However, it is advisable to be

cautious as regards the absolute accuracy of the results, inaccuracies in the frequencies producing an effect the importance of which becomes larger and larger as the time increases.

Table I.

Power as to the disturbing masses I

E v o l u t i o n i n the Computation o f Long-TermVariationa o f the E a r t h ' s O r b i t a l Elemants

Degree in planetary eccentricities and inclinations I

Lagrange (1781) Laplace (I 798) Pont~coulant-Le Verrler (1834) I Stockwell (1870) Harzer (1895) I Milankovitch-Stockwell-Pilgrim (1920) I Milankovich-Miskovitch-Le Verrier (1941) Anolik I (1969) Bretagnon 1 (1972-1974)

2 2

Brouwer-Van Woerkom (1950) Sharaf-Boudnikova (1969) Vernekar (1972)

3

Anolik e t a~. (1969) Bretagnon 2 (1974)

Bretagnon 3 (1974) I BerBer I (1973-1977a) 2 BerBer 2 (1976) 2 BerBer 3 (1978a)

Contributions to the long-term variations of the Earth's orbital elements have been classified according to the accuracy of the series expansions used for the eccentricity and inclination systems (i.e. following the degree in planetary es and is and the power as to the disturbin B masses), and also according to the degree of the series expansion for the obliquity and the annual general precession in longitude (number given in front of the author's name). The larger the numbers are, the larger is the number of terms kept in the series expansions and, thus, the higher is the accuracy. As an example, the upper two groups lead generally to a solution which is accurate only for lO0,000 years from today. More detailed information is given in BerBer (1977a) (Table updated from BerBer (1978b)). As S O has been taken as a constant, 1353 W/m 2 (! .95 cal/cm2/min), and a has no purely secular part when the perturbations of the second order are included, only the long-term variations of e, c and ~ must be determined.

This solution (labelled BerBer 3 in Table I) for the

classical astro-insolation parameters can be written in a simple practical form, still providing excellent accuracy (r_he most important hiBher-order terms are included in the series expansions) :

The Milankovitch Astronomical Theory of Paleoclimates: A Modern Review

e = e 0 + E E i cos (Ait + ~i )

(I)

e s.~n ~ = EP i sin (~i t + 8i) EA i

¢ = ~' +

cos

(yi t

+

I09

(2) 61).

(3)

All the constants in (I), (2) and (3) are available in Berger (1978a).

Numerical experiments conducted over the previous 5 million years show that e varies between 0.0005 and 0.0607 (present value 0.0167), with an average quasi-period of 95,000 years, and that ¢ varies between 22°2 ' and 24°30 ' (today's value 23°27'), with a quasi-perlod of 41,000 years.

The revolution of the vernal ~oint relative to the moving perihelion (climatic

precession) has an average quasi-period of 21,700 years (e sin ~ is presently equal to 0.01635), whereas relative to the fixed perihelion of reference, this quasi-period is 25,700 years (astronomical precession of equinoxes).

The p r o c e d u r e used f o r o b t a i n i n g l o n g - r a n g e c h a n g e s i n the e l e m e n t s o f p l a n e t a r y o r b i t s involves a drastic

simplification

o f t h e a s t r o n o m i c a l p r o b l e m and the r e s u l t s

admittedly of a limited accuracy.

The most s e r i o u s l i m i t a t i o n s

from t h e u r ~ e r t a i n t i e s

of the periods themselves.

m a s s e s and t h e o r b i t a l

elements of the p l a n e t s .

p e r i o d s s h o u l d n e e d a c o r r e c t i o n o f 1%, i t

estimate,

I f any one o f t h e p r i n c i p a l

i s c l e a r t h a t t h e c o n t r l b u e ~ o n due t o t h i s

t h u s becomes l e s s and l e s s r e l i a b l e

from the e p o c h i n c r e a s e s .

100 p e r i o d s .

as t o d e t a i l

The r e p r e s e n t a t i o n

as the i n t e r v a l

o f time

I f au u n c e r t a i n t y o f 1% i n a p e r i o d o f 90,000 y e a r s i s a r e a s o n a b l e

t h a n t h e r e p r e s e n t a t i o n by t h e s e s e r i e s h a s l o s t a l l i t s meaning a s t o d e t a i l

9 x 106 y e a r s . the calculations

in

Moreover, due to t h e l a r g e i n f l u e n c e o f the J u p i t e r - S a t u r n n e a r - r e s o n a n c e , most b e r e p e a t e d i n c l u d i n g :

(i) not only l o n g - p e r i o d terms of f i f t h

( t e r m s w h i c h would a f f e c t m o s t l y Mercury, Venus, E a r t h and M a r s ) , b u t a l s o , planets,

nature arise

These a r e o b t a i n e d as f u n c t i o n s o f t h e

t e r m would b e a f u l l p e r i o d o u t o f p h a s e a f t e r

by t h e s e s e r i e s

obtained are

They would be m o d i f i e d by t h e i n c l u s i o n o f

t e r m s o f h i g h e r o r d e r i n m a s s e s and h i g h e r powers i n e s and i s .

particular

of a practical

short-period

terms o f h i g h e r o r d e r i n e s and i s ,

third order in masses, at least

and e v e n t u a l l y ,

order

(2) f o r a l l

the

(3) t e r m s o f the

for the Jupiter-Saturn near-Tesonanoe.

However, qualitative and quantitative indications are also available to show r_hat the results are nevertheless reliable: (I)

The same method has been used with good success (Brouwer,

1953) in the treatment of

satellite systems in which the periods may b e shorter by factors of about I0~. 100 years in such a satellim

Thus,

system may correspond to a million years in the planetary

system. (2)

The application of the Brouwer-van Woerkom solution to the morion of minor planets indicates a general reliability to about the third decimal place, but does not indicate anything concerning the interval of time for which the de~lopments may be trusted.

(3)

There is an excellent agreement between the mean long-term period of ¢ and that computed from a numerical expression derived from the last 2000 years of observation (Wit tlnan, ]979).

To sum up, some improvements are now expected in the field of theoretical researches, in the techniques used to determine the planetary motion and in the number of terms kept in the expansions used.

The actual representation of the elements of the Earth's orbit in the

simple trigonometrical form may be considered to be reliable in detail to four siEnificant

A. B e r g e r

I10

figures for approximately 3 million years.

Even if, on account of the uncertaintT of the

periods involved, the details of the representation become more and more uncertain, the farther the epoch to be considered is removed from the present, the general character of the representation remains the same even for very distant epochs.

Although the problem of cross-dating is even more complicated due to the existence of the same kind of timing uncertainty in geological data, the overall accuracy in paleoclimatology is now improving and an absolute chronology of paleoclimates over the last 3 million years may be attempted.

3.

THE MILANKOVITCH THEORY*

Adhemar (1842) was t h e f i r s t variations

t o s u g g e s t t h a t the prime mover o f t h e i c e ages m i g h t b e

i n t h e way t h e E a r t h moves around t h e Sun.

over long p e r i o d s of time, v a r i a t i o n s C r o l l (1875) climate.

He b a s e d h i s t h e o r y on the f a c t t h a t

occur in the direction

of the E a r t h ' s a x i s .

took Adhemar's i d e a s and d e v e l o p e d them i n t o a new a s t r o n o m i c a l t h e o r y o f

He r e a s o n e d t h a t a d e c r e a s e i n the amount o f s u n l i g h t r e c e i v e d d u r i n g t h e w i n t e r

f a v o u r s t h e a c c u m u l a t i o n o f snow, and t h a t any s m a l l i n i t i a l

increase in the size of the area

c o v e r e d by sr~w would b e a m p l i f i e d by t h e s n o w f i e l d s t h e m s e l v e s ( p o s i t i v e having deter~ned

feedback).

After

which a s t r o n o m i c a l f a c t o r s c o n t r o l t h e amount o f s u n l i g h t r e c e i v e d d u r i n g

the w i n t e r , h e c o n c l u d e d t h a t t h e p r e c e s s i o n o f t h e e q u i n o x e s must p l a y a d e c i s i v e r o l e . a l s o showed t h a t c h a n g e s i n t h e s h a p e o f the o r b i t wobble i s i n c h a n g i n g the i n t e n s i t y

d e t e r m i n e how e f f e c t i v e

o f the seasons.

Croll's

first

He

the p r e c e s s i o n a l

theory predicts

t h a t one

h e m i s p h e r e o r the o t h e r w i l l e x p e r i e n c e an i c e age w h e n e v e r two c o n d i t i o n s o c c u r s i m u l t a n e o u s l y : a markedly e l o n g a t e d o r b i t Sun.

and a w i n t e r s o l s t i c e

t h a t o c c u r s f a r from thp.

L a t e r o n , C r o l l h y p o t h e s i z e d t h a t an i c e age would be more l i k e l y

to o c c u r d u r i n g

periods when the axis is closer to vertical, for then the polar regions receive a smaller amount of heat.

As time went on, many geologists in Europe and America became more and more dissatisfied with troll's theory, finding it at variance with new evidence that the last ice age had ended not 80,000 (according to his view) but I0,000 years ago.

Moreover, theoretical arguments were

advanced against the theory by meteorologists, who calculated that the variations in solar heating described by Croll were too small to have any noticeable effect on climate.

I t was o n l y d u r i n g the f i r s t Spitaler

d e c a d e s o f the 20th c e n t u r y t h a t K~ppen and Wegener (1924),

(1943) and above a l l M i l a n k o v i t c h (1920, .1930,

1941) i n d i c a t e d

that it

is the

d i m i n u t i o n o f h e a t d u r i n g the summer h a l f - y e a r which i s the d e c i s i v e f a c t o r i n g l a c i a t i o n . In f a c t ,

this theory requires

that a northern high-latitude

s u n d e r must b e c o l d to p r e v e n t

t h e w i n t e r snow from m o l t i n g i n s u c h a way as to a l l 0 w a p o s i t i v e v a l u e i n t h e annual b u d g e t o f snow and i c e , and to i n i t i a t e

a positive

f e e d b a c k c o o l i n g o v e r t h e E a r t h d~rough a f u r t h e r

e x t e n s i o n o f the snow c o v e r and s u b s e q u e n t i n c r e a s e o f t h e s u r f a c e a l b e d o . of a perfectly

On t h e a s s u m p t i o n

t r a n s p a r e n t a t m o s p h e r e , t h a t h y p o t h e s i s t h u s r e q u i r e s a minimum i n t h e

n o r t h e r n h e m i s p h e r e summer i n s o l a t i o n

at high latitudes.

*A very complete stmmmry of historical works on astronomical theory is available in Imbrie and Imbrie (1978), and a detailed llst of references is given in Berger (1978c).

The Milankovitch Astronomical Theory of Paleoclimates: A Modern Review

In fact,

a caloric

astronomical

half-year

comprises all

(l)

was i n t r o d u c e d by M i l a u k o v i t c h , b e c a u s e t h e l e n g t h o f the

s e a s o n s shows s e c u l a r v a r i a t i o n t h e days o f s t r o n g e r

mathematical definition

111

(the caloric

radiation;

of such intervals

the longitude of the perihelion

summer i s t h e h a l f - y e a r

t h e o t h e r one b e i n g t h e c a l o r i c

implies that glaciation

is such that

which

winter).

The

w i l l o c c u r when:

the northern hemisphere stmner begins at

the aphelion (referring to Fig. 3, this means ~ - 270 ° ) ; (2)

the eccentricity is maximom, which means that the Earth-Sun distance at the aphelion will be the largest.

This eccentricity affects the relative intensity and the duration

of the seasons in the different hemispheres, but also the difference between maximum and minimum insolation received in the course of ] year, a difference which can amount to as much as 30% for the most elliptical orbit. the climatic precession parameter esin ~.

In fact, e and ~ are mainly used through

This parameter plays an opposite role in

both hemispheres, and is a measure of the difference in length between half-year astronomical seasons and of the difference between the Earth-Sun distance at both solstices.

It is the most important factor because critical climatic parameters such

as insolation at solstices and caloric season insolation require its numerical values (Berger, (3)

1978b)

obliquity is low, which means that the difference between stnmner and winter is weak and the latitudinal contrast is large.

Following all latitudes

these requirements,

n o t o n l y would t h e summer t e m p e r a t u r e s i n n o r t h e r n h i g h

b e f r e s h enough to p r e v e n t snow and i c e from m e l t i n g , b u t a l s o m i l d w i n t e r s would

allow a substantial and p o l a r l a t i t u d e s ,

evaporation in the intertropical

the humidity being supplied there by an intensified

due t o a maximum l a t i t u d i n a l

general circulation

"thermal" gradient.

Moreover, as t h e h i g h - l a t i t u d e o f low l a t i t u d e s

zone and a b u n d a n t s n o w f a l l s i n t e m p e r a t e

are essentially

caloric

insolations

a r e m a i n l y d e p e n d e n t o n ¢, w h e r e a s t h o s e

d e p e n d e n t on e s i n ~, and a s t h e c - e f f e c t

i s t h e same i n

both hemispheres, whereas the ealn ~ effect is opposite, the nature itself of this model

i m p l i e s c o m p e n s a t i o n o f n e g a t i v e summer d e v i a t i o n s

by p o a i t l v e

a n t i s y m m e t r y b e t w e e n h e m i s p h e r e s w h i c h becomes m i n i m a l f o r a l l

4.

Till

winter deviations latitudes

and a n

h i g h e r t h e n 70 ° .

MODERN VERSION OF THE MILANKOVITCH THEORY

a r o u n d 1973, t h i s t h e o r y h a s b e e n l a r g e l y d i s p u t e d b e c a u s e t h e d i s c u s s i o n s

on f r a g m e n t a r y g e o l o g i c a l 65 ° n o r t h , in conflict

were b a s e d

s e d i m e n t a r y r e c o r d s and on M i l a n k o v i t c h s , - ~ , p r r a d i a t i o n

t h e a b s o l u t e a c c u r a c y o f which was n o t d e m o n s t r a t e d . w i t h some w e l l - a d m i t t e d

observations,

Moreover, t h i s

-=-~ly the quasi-simultaneity

curves at

t h e o r y was of glacial

ages in both hemispheres.

Despite improvements in dating and in interpretation of the geological data in terms of

paleoclimates

b y E m i l i a n i i n 1955, and t h e use o f some o t h e r i n s o l a t i o n

c u r v e s by E r o e c k e r

in 1966, the following four fundamental questions related to the Milankovitch theory were not answered yet: (I)

Are the long-term variations of the Earth's orbital elements and of the insolation

(2)

Are the quasi-periodicities of the Earth's orbital elements significantly present in

reliable?

the geological records?

112

A. Berger

(3)

Is there any significant correlation between insolation curves and geological data?

(4)

Can these insolation changes have induced climatic changes of a magnitude similar to those which have been recorded in the Reo-ecological data?

4.1.

Spectrum ° f As.tronomical and Geological Data

Although a new modern solution has been produced in 1973 by Berger, it was only in 1976 that Hays et u~., and Berger, demonstrated that the astronomical frequencies were significantly present in paleoclimatic data.

Indeed, from a careful spectral analysis of records in

various deep-sea cores, Hays et GI. (1976) have shown that the following quasi-periods were statistically significant: I05,000, 41,000, 23,000, and 14,000 years.

At the same time,

Berger (1977b) was determining all parameters of equations (1) to (3) where the periods related to the most important terms were proving to be respectively: for the eccentricity, 413,000, 95,000, 123,000 and I00,000; for the obliquity, 41,000; and for the paleoclimatic precessional term, 23,700, 25,400, 18,980 and 19,160; for these three elements, secondary peaks appear to be located respectively around 50,000, 53,000 and 30,000, 15,000 and 56,000 years.

S i m i l a r p e r i o d i c i t i e s have already been found i n quaternary g e o l o g i c a l data by Chappell i n 1973 and, maybe i n a less rigorous way, by Mann (1967) and Van den Heuvel (1966b), and even i n o l d e r epochs by Bradley (1929), Van Houtan (1,964) and Bond and Stocklmayer (1967). This is remarkable, even if the time scale used in these investigations differs from Hays et ul. by about a factor of two.

The explanation of this paradox is to be found in the

relative value of the astronomical periodicities and in the dominant climatic periodicity which in all of the Hays et ul. cores is the lO0,000-year cycle, and not, as expected, the geological response to the 41,000-year obliquity and to the 21,000-year precession cycle as predicted by a linear version of the theory of orbital control.

Indeed, the spectral peak

identified by Chappell as due to precession and by Van den Heuvel as due to precession half-cycle are now to be understood as the effect of, respectively, the obliquity and a full precession cycle.

In a recent publication, Kominz (1978) has shown that records from deep-sea core V28-239 display spectral peak periodicities centered on |04,000, 92,300, 58,500, 52,200, 41,000, 30,000, 23,000 and 19,000 years.

In order to be sure that this observation was not simply

an artifact of visual curve matching, the coherency between the ~180 records and the orbital variations has been determined by cross-spectral analysis.

Significant peaks with a

coherency greater than 0.40 are only related to precession and obliquity, with obliquity consistently leading the ~180 record by about lO,000 years.

For the eccentricity, the situation is far more complicated.

The most important term of a

412,000-year period, needing a time series record ions enough to distinguish such a periodicity, has been found in geological records only quite recently (Kominz, 1978; Harrell and Briskin, 1978; Shackleton, ]978), even if it has been forecast as early as 1976 by Berger.

Moreover, its interpretation, and that of peaks in the range of |O0,000-year

containing most of the climatic variance, is difficult.

These quasi-perlods may be related

either to the eccentricity periods themselves, or to a beat effect due to the non-linear interaction between the two precession peaks (Wigley, 1976).

The problem is still unsolved

The Milankovitch Astronomical Theory of Paleoclimates: A Modern Review

b e c a u s e we have to f i n d how much v a r i a n c e o f t h e c l i m a t i c 100,000-year period which originates from t h e n o n - l i n e a r (Birchfield

directly

other astronomical variables

among l a t i t u d e s . direction

and how much o r i g i n a t e s

system to the precessional

forcing

Looking t o t h e i n f l u e n c e o f e a l o n e on t h e i n s o l a t i o n ,

can see that it acts only through the (1-e2) "1/2 factor Earth, all

c a n b e e x p l a i n e d by t h e

from t h e e c c e n t r i c i t y

response of the climatic

and Weertmau, 1978).

variations

113

contributing

Although these variations

a c c o r d i n g to t h e r e s u l t s

in the total

e n e r g y r e c e i v e d by t h e

to a redistribution

a r e s m a l l (~ 0 . I Z ) ,

we

of the energy

they are in the right

o f Hays e# u l . : t h e y a r e p o s i t i v e l y

correlated

with the

change i n sumner t e m p e r a t u r e and n o t i n t h e o p p o s i t e s e n s e as r e q u i r e d b y M i l a n k o v i t c h , who stated that. the influence of e was almost exclusively through the esin ~ term.

However, t h e c o h e r e n c y p e a k s t h a t may b e a s c r i b e d do n o t g a l l d i r e c t l y

to eccentricity

on t h e 400 and 1 0 0 x l 0 3 - y e a r g e o l o g i c a l p e r i o d i c i t i e s .

these spectral

p e a k s do n o t b e a r a d i r e c t

Earth's

and t h a t t h e 26Z o f t h e v a r i a n c e r e c o r d a t t r i b u t e d

orbit,

is only attributed

to p r e c e s s i o n

form t h e o n l y d e t e r m i n i s t i c tested,

(Kominz and P i s i a s ,

linear

relationship

(6Z) and t o o b l i q u i t y

component o f the c l i m a t i c

i n n a t u r e end c a n b e

model o f H a s s e l m a n ( 1 9 7 6 ) , i n w h i c h t h e ~180 r e c o r d i s f o r c e d b y t h a n 12,000 y e a r s .

A l t h o u g h t h i s v a l u e o f 26Z i s math weaker t h a n t h e 80g f o u n d b y Hays e t a l . t h e v a r i a n c e i n t h e p e a k s i n t h e power s p e c t r u m o f t h e c l i m a t i c

attributed

to astronomical

forcing,

( 1 9 7 3 ) , who d e s c r i b e d c l i m a t e change as a n a l m o s t p e r i o d i c

(as i n B i r c h f l e l d

and W e e r ~ a n ,

Insolation

and S e n s i t i v e

Most o f the s c i e n t i s t s glacial

oscillations

assuming

record could be

stochastic

process,

1978) t o t e s t

it

However, i f mid-mouth i n s o l a t i o n

elements or even instead of the M/lsnkovitch caloric

is found for the explained variance

4.2.

(1976),

and i s e v e n w e a k e r t h a n t h e 41Z o f Chin and Y e v j e v i c h

h i g h e r t h a t t h e ]0Z s u g g e s t e d b y Keer ( 1 9 7 8 ) . instead of these orbital

forcing

parameters

record in the frequency range being

a much h i g h e r f r e q u e n c y p r o c e s s c o r r e s p o n d i n g t o a p e r i o d o f l e s s

that all

that

of the

to astronomical

If these orbital

t h e n t h e r e m a i n i n g n o n - r a n d o m v a r i a n c e must b e s t o c h a s t i c

e x p l a i n e d by t h e s t o c h a s t i c

This implies

to the eccentricity

(20Z).

1979)

i s much

i s used insolatluns

t h e a s t r o n o m i c a l t h e o r y , a much h i g h e r v a l u e

(up to 87Z i n B e r g e r e t a Z . ,

1980).

Latitudes

who u s e d t h e M i l a n k o v i t c h a s t r o n o m i c a l during the Quaternary based their

g e o l o g i c a l c u r v e s and s , - , , ~ r i n s o l a t i o n

theory to account for the

c o n c l u s i o n s on a m a t c h i n g b e t w e e n

c u r v e s o b t a i n e d by M i l a n k o v i t c h ( ] 9 4 1 ) ,

Van Woerkom ( 1 9 5 3 ) , B e r n a r d ( 1 9 6 2 ) , o r most r e c e n t l y by V e r n e k a r (1972) and B e r g e r ( 1 9 7 8 b ) . The f i r s t

criterion

which has been used is a visual or statistical

minima and maxima o f b o t h c u r v e s (Brouwer, radiation

19501 J a r d e t s k y ,

1961).

correspondence between M i l a n k o v i t c h s,-m-pr

c u r v e s f o r 65°N h a v e b e e n u s e d more f r e q u e n t l y b e c a u s e o f t h e more e x t e n s i v e

nature of Pleistocene

glaciation

in the northern hemisphere.

of the Milankovitch school are Emillani-Ceiss using insolation

c h a n g e s to s t a r t

( n o t to e n d ) g l a c i a l

suggested that the Scandinavian ice sheet had its A n a l y s i s o f the i d e a l l z e d

Among t h e most a c t i v e p a r t i s a n s

( 1 9 5 7 ) , who p r o p o s e d a t h e o r y o f g l a c i a t i o n cycles,

origins

a n d Z e u n e r ( 1 9 5 9 ) , who

at approximately that latitude.

c u r v e f o r t e m p e r a t u r e s o f t h e s u r f a c e w~cers o f t h e A t l a n t i c

g i v e n by E m i l i a n i l e d B r o e c k e r (1966) t o s u g g e s t a n a s t r o n o m i c a l - c l i m a t i c original

combination of precession,

tilt

and e c c e n t r i c i t y .

The o v e r a l l

Ocean

c u r v e b a s e d upon an effect

of his weighting

A. B e r g e r

114

factors was to create a deep 18,000 YBP minimum, two important maxima at 120,000 and 80,000 YBP, and a long period of intermediate climate between 70,000 and 22,000 BP, where the 48,000 YBP prominent warm peak found in the Milankovitch curve is greatly depressed.

Assumln 8

the half-response time for the continental glaciers to be 3000 years, his ice-volume curve became consistent with observation.

Following this idea, Broecker e~ ul. (1968) used the 45°N

insolation curve (where the precession effect is given more weight than is tilt effect) in such a way that the warm peak at 50,000 YBP is largely removed and a new peak appears at I06,000 YBP.

These results clearly indicated that the last four sea-level high stands

(122, 103, 82, 5000 YBP7 correspond closely in time to the last four prominent warm peaks (127, 106, 82, ii,000 YBP) in the modified curve of sunder insolation, not only in chronology but also in magnitude (Mesolella e~ ul., |9697.

However, based on very accurate dating of

core V12-122, Broecker and van Donck (1970) had to increase by 35% the absolute time scale adopted by Em/liani for deep-sea cores.

They then proposed 45, 55 and 65°N sunmmer insolation

as a tool to explain the gradual glacial build-ups over periods averaging 90,000 years in length and terminated by deglaciations completed in less than one-tenth of ~his time.

At the

same time, Veeh and Chappell (1970) obtained for the last 230,000 years the same kind of correlation between sea-levels derived from raised coral reefs of New Guinea and summer insolation at 45°N.

~hese qualitative coincidences of the principal maxima and minima of both curves are, however, somewhat illusory because they are not based on purely objective analysis but most of the time merely on preconceived ideas.

Ruddiman and McIntyre (19767 have proposed the insolation

at 55°N as a guideline, and van den Heuvel (1966a) that at latitudes norwh of 70°N because there the asynznetry between northern and southern hemispheres is largely attenuated, due to the fact that in Arctic and Antartic regions, the insolation variations caused by changes of the obliquity are much larger than those caused by changes in the precessional parameter. Following the same idea, Evans (19727 built up an absolute time scale for the whole Pleistocene from insolation received both at 65°N and 65°8, making an allowance for latent heat, albedo and 4000 years time lag.

This is in opposition to Falrbridge (|961), who has

referred to the northern hemisphere middle latitudes (40-75")

as the sensitive latitudes to

climatic change, since approximately 95% of the world's mountain glaciers are located at this position.

Kukla's (19727 proposal is based on the season-to-season difference in the

mean insolation received beyond the latitudinal belt lying bet~een 25 and 75°N in the winter half-years of the northern hemisphere,

the fastest increase of this quantity being supposed

to indicate interglacials such as Eem and Holocene.

The m o d e l s b y C a l d e r a decline

(17

(1974)

i n summer s u n s h i n e

cal/cm2/day) allows

and Mason (1976) at

are even more complete.

50°N (47 c a l / c m / d a y )

below a certain

Calder

supposed that

level

the volume of glaciers and ice sheets to grow in simple proportion

to the deficit, while st~-~,m~rsunshine above that level melts ice with a different proportionality.

With a melting rate five times the freezing rate, a realistic curve for the

most recent glaciation (the last 78,000 years) has been obtained and applied over the past 860,000 years.

Mason has reworked some aspects of the Milankovitch model in terms of the

variations in the amount of heat received each year north of 45°N, a variation which is about I% of the total heat received by the polar cap.

When ice cover developed during the period

from 83,000 to 18,000 YBP, the integrated deficiency in such insolation amounted to some 556 cal/g of ice formed.

From 18,000 to 6000 YBP (melting of the ice), the integrated

surplus of heat received was 1025 cal, while the latent heat required for the known decrease in volume of ice was 3.2×102~ cal.

Such a close agreement with the corresponding latent

The M/lankovltch Astronomical Theory of Paleoclimates: A ~odern Review|

heats might not be a pure coincidence.

115

This simple astronomical model shows that the

northern hemisphere insolatlonminimumwill

be reached in about 10,000 years but

bottoming-out before reaching conditions quite so extreme as those which prevailed at the height of the recent ice age.

However, more o b j e c t i v e principal

techniques of data analysis,

component a n a l y s i s ,

between orbital

and s p e c t r a l

e l e m e n t s and g e o l o g i c a l

analysis

data.

simulates

the relative

amplitude of gross paleoclimat~c indicators

=

regression,

The following astro-climatic

successfully

(Kukla and B e r g a r ,

the timing within

such a s m u l t i v a r i a t e

c o u l d b e used t o model t h e r e l a t i o n s h i p

the precision

limits

i n d e x (ACLIN)

of the radiometric

during the last

d a t i n g and

250,000 y e a r s

]979):

~ - 9018°I~

÷ c' - 22 + 500

e2

where ~ and e are orbital elements at time 4, and ¢' is the obliquity at a time ~' separated by the angular measure of 90" in the longitude of perihelion (approximately 5000- 6000 years). ACLIN predicts the interglaclals at 6 and 122 x 103 YBP, the interstatials 28, 54, 80 and |0l x 103 I'BP, and the cold maxima at around ]8, 66, 89 and l]! x 103 l~P, with the last one lass firmly expressed than the first two.

In addition

to all

synchzonlcit7

these proofs,

of northern

the remaining naming

and s o u t h e r n g l a c i a t i o n s

The e v i d e n c e i n d e e d c o n f i r m s t h a t b o t h h ~ s p h e r e s with an indication

that

t h e 18,000 YBP c a s e . present

distribution

sea --provides because it of ice.

Indued~

(Gribbin,

1978).

undergo n n a r - s y n c h r o n o u s g l a c i a t i o n s ,

of continents cold northern

for

i n t h e n o r t h e r n h e m i s p h e r e Bet " c o l d e r " when t h o s e i n the

t h i s e v i d e n c e was l o n g t a k e n as r e f u t i n g

the model.

However, t h e

-- s o u t h e r n p o l a r c o n t i n e n t and a l a n d - l o c k e d n o r t h e r n p o l a r

t h e c l u e to e x p l a i n

requires

also disappeared

t h e s o u t h l e a d s t h e n o r t h i n t o and o u t o f an i c e aBe, a t l e a s t

Since sm~ers

s o u t h e r n become ' ~ a r m e r " ,

d o u b t a b o u t t h e mechanism -- t h e n e a r

--has

global ice-ages within

the overall

smmners and c o l d s o u t h e r n w i n t e r s

M i l a n k o v i t c h models

to i n i t i a t e

the spread

in qualitative terms, it is clear that this initiation in the northern

hemisphere requires cool s,l-wers to prevent winter snow and ice from melting.

On the other

hand, in the southern hemisphere, there is scarcely any land at high latitudes which is not already covered by permanent ice caps.

Snow which falls on the sea cannot remain even if

s,-w,~ers are cool, rand the only way to spread the covering of sea-ice is to have very severe winters in which large volumes of ocean water are frozen, when once again the increased albedo plays a part in hastening the development.

4.3.

Monthly Insolation and Paleoclimates

It is not only the sensitive latitudes which play an important role in climate modelling, but also the critical season.

In that way, some important conclusions may be drawn from

the fact that the obliquity peak in geological records is smaller than the lO0,000-year peak or, at least, less dominant than formerly thought (Bernard,

1962).

By analysing the solar

radiation available on the assumption of a perfectly transparent atmosphere (Berger,

1975),

it can be deduced, on the one hand: (])

the insolation received at the equinoxes and the differences between the length of t_he s,-~-er and of the winter astronomical seasons are functions only of the precessional parameter;

A. B e r g e r

I16

(2)

at the solstices, this parameter has a larger influence than the obliquity.

On the other hand: (1)

during astronomical seasons, the insolation received at any latitude is a function of

(2)

at high lati tudes, the deviations of the insolation from present-day values for the

the obliquity ¢, and

caloric seasons are mostly functions of ¢.

Moreover, t h e M i l a n k o v i t c h c a l o r i c variations,

and t h e whole s u ~ r

half-years

y e a r t o e x p l a i n t h e advance and r e t r e a t consequence,

mask t h e i n t r a - a n n u a l

season is not necessarily

o f i c e s h e e t s and g l a c i e r s

the simulation of the past climate will

daily or monthly insolation

variability

t h e most s e n s i t i v e (Kukla,

and i t s period of the

1975).

As a

t h u s need t h e knowledge o f t h e p a s t

instead of, or in addition

to,

the Milankovitch caloric

season

insolation. As a n i U u a t r a t i o n ,

it could be very interesting

this monthly insolation Milankovitch caloric

to show f i r s t

how t h e n u m e r i c a l v a l u e s o f

pattern

(Berger,

1978a) compare w i t h and comple-~'nt t h e c l a s s i c a l

insolation

(Berger,

1978b).

For example,

18,000 y e a r s ago when a g l a c i a l

maximum o c c u r r e d w i t h a n A r c t i c i c e - c a p e x t e n d i n g t o t h e N e t h e r l a n d s , New York, t h e o b l i q u i t y , respectively insolatlon

the precessional

2 3 " 2 7 ' , 0.00544 and 0 . 0 1 9 4 5 , close

to t h e i r

lead to an annual insolation

present-day valua.

( F i g . 4) f o r t h e 60-70"N b e l t

and a c a l o r i c

s,,-,-~r

anomalies

amounted t o - 3 5 c a l / c m 2 / d a y i n A u g u s t - S e p t e m b e r and

corresponding daily ineolation.

respectively

8 a n d 5% o f t h e a c t u a l

S i m i l a r c o m p a r i s o n s made o v e r t h e l a s t

of the caloric

5%, w h e r e a s t h e d e v i a t i o n s

E n g l a n d and

whose v a l u e s were

However, t h e m o n t h l y i n s o l a t i o n

*40 c a l / c m 2 / d a y i n A p r i l - M a y , which r e p r e s e n t that the deviations

central

p a r a m e t e r and t h e e c c e n t r i c i t y ,

ineolations

from t h e i r

for the monthly insolations

1,000,O00 y e a r s show

p r e s e n t - d a y v a l u e s a r e always b e l o w

sometimes r e a c h 12% ( B e r g e r ,

1979b).

I00.

18,000 78P ,d )- 50-

5

:S

t) .J < t)

O

o

(3 u

:';,

\,

/

k..., -~o

-IOC

Fig. 4.

Annual cycle of insolation their 1950 A.D. values, of zonal belt. The full li-~ (cal/cm2/d) and the dashed of 1950 A.D. values).

18,000 Y.B.P. Deviations, from zonal monthly mean for 60-70°N is related to the left scale line to the right scale (%age

Now, t o f i g u r e o u t how t h e c h a n g e s i n t h e a n n u a l c y c l e from one 1 0 0 0 - y e a r p e r i o d t o t h e following one are related to climatic variations, the dynamic behaviour of some insolation features must be analysed (Berger,

1979b).

Many times during the last million years, a

maximum in May shifts progressively towards July-August at the same time r-hat a mini-,,m appears in February and moves towards April.

From time to time, this shift is faster, the

deepening in the spring season is steeper, and is then rapidly replaced by a maximum, the

The M i l a n k o v i t c h A s t r o n o m i c a l Theory of P a l e o c l i m a t e s :

spring minimum being shifted towards summer.

A Modern Eeview

117

This feature, called insolation signature, is

thought to be related to a warm phase going into a cool one, a shortening of the lag times for the oceanic and cryospheric responses being expected due to the power of such insolation

changes. Over t h e last 500,000 y e a r s ,

such i n s o l a t i o n

signatures

243, 220, 199, 127, 105, 84 and 13 t h o u s a n d YBP. i n l a t e w i n t e r and e a r l y s p r i n g .

start

a r o u n d 505, 486, 465, 315, 290,

At t h e s e d a t e s ,

a weak minimum i s o b s e r v e d

A s t r o n g maximum c u l m i n a t e s i n J u n e - J u l y and i s sometimes

r e p l a c e d by a deep minimum i n some 7000 y e a r s . w i t h t h e maxima o f t h e g e o l o g i c a l

I t i s r e m a r k a b l e how w e l l t h e s e d a t e s f i t

c u r v e , namely t h a t deduced from d e e p - s e a c o r e HC13-229

whose time s c a l e h a s r e c e n t l y b e e n r e v i s e d ,

t h e b o u n d a r i e s from c o l d to warm main s t a g e s

b e i n g now l o c a t e d 504, 422, 335, 245, 128 and lO t h o u s a n d YBP (Morley and Hays, This analysis calculated

of the deviations

over the last million

of the insolation climatic

years,

clearly

indicates

preferably

from t h e mean s t a t e

that variations

in the annual cycle

and i n t h e a m p l i t u d e o f t h e m o n t h l y i n s o l a t i o n

changes.

steadiness glacial

o f mid-month d a i l y i n s o l a t i o n ,

It is not a linear real-time

relationship,

i n time which i s t h o u g h t t o b e r e s p o n s i b l e

or interglacial.

Accordingly,

the very flat

60*N b e t w e e n 7 0 , 0 0 0 and 3 0 , 0 0 0 YBP must i n d i c a t e a r o u n d 75,000 YBP.

The deep minimum s t a r t i n g

degree of

f o r the a p p e a r a n c e o r n o t o f a f u l l pattern

of the annual cycle observed at

t h e m a i n t e n a n c e o f the t r i g g e r e d

glaciation

i n A p r i l a r o u n d 30,000 YBP and moving towards for the last

glacial

maximum

g o i n g from a l a r g e maximum summer i n s o l a t i o n

deep minimum and b a c k t o a maximum i n a v e r y l i m i t e d h a s to be e x p e c t e d .

to Q u a t e r n a r y

is their

When t h e s p e e d o f change o f t h e a n n u a l c y c l e p a t t e r n

from one 1 0 0 0 - y e a r s t o t h e n e x t i s f a s t , glaciation

are related

but it

J u n e and J u l y b e t w e e n 25,000 and 20,000 YBP must b e r e s p o n s i b l e t h a t o c c u r r e d a r o u n d 2 0 - 1 8 , 0 0 0 YBP.

1979).

to a

time span, a short cooling or "abortive"

I t i s t h e c a s e from 127,000 (maximum) t o 118,000 YBP (minimum)

and t o 107,000 YBP (maximum again). As f a r a s t h e p r e s e n t Holocene i n t e r g l a c i a l maximum i s c l e a r l y the present-day reach its

related

insolation

to its

beginning.

is concerned, From o r b i t a l

t h e 11,000 YBP J u l y i n s o l a t i o n geometry, it

is also evident that

d u r i n g stumner months h a s b e e n d e c r e a s i n g s i n c e 3000 YBP, w i l l

minimum i n J u l y - A u g u s t a r o u n d 3000 YAP, and w i l l

not be significantly

larger

than

t h e mean s t a t e b e f o r e 24,000 YAP. Finally, state

as a t 9 4 , 0 0 0 ,

105,000 and 197,000 YBP, t h e main m e x i - - - , d e v i a t i o n

from t h e mean

i s l o c a t e d a t t h e s o u t h and n o t t h e n o r t h p o l e , as f o r t h e o t h e r r e l e v a n t

dates;

it

becomes e v i d e n t t h a t t h e n o r t h e r n h e m i s p h e r e s h o u l d n o t always b e c o n s i d e r e d a t t h e l e a d i n g hemisphere.

Consequently,

t h e p r o b l e m o f l a g s and l e a d s a t t h e Q u a t e r n a r y time s c a l e

b e t w e e n t h e n o r t h e r n h e m i s p h e r e and t h e s o u t h e r n h e m i s p h e r e (CLIMAP, ]976,1978) c a n p r o b a b l y be partly

4.4.

resolved through the analysls

o f t h e mid-month i n s o l a t i o n

behaviour in the past.

M o d e l l i n g t h e Dynamics o f C l i m a t i c Changes

~he M i l a ~ k o v i t o h model h a s t h u s p a s s e d b o t h t h e t e s t o f p h y s i c a l

plausibility

and s e v e r e

statistical tests, but there still remain difficulties in explaining how the relatively smell changes in Milankovitch insolation could be sufficient to initiate or end glacial ages.

l]~

A. B e r g e r

In models by Shaw and Donn (1968) and Sellers (1970) which, under astronomical forcing, generate climatic variations one order of magnitude too smell, the albedo feedback mechanisms in high latitudes, the oceanic circulation, and the ocean-climate interactions (Barnett, 1978) were poorly simulated.

If the seasonal extent of ice cover (Veeh and Chappell, 1970) and

non-linear feedbacks related to the polar ice caps (Adam, 1975; Fredsriksen, 1976), are included correctly, results begin to be much more significant (Johnson and McClure, 1976; Weertman, 1976).

In this respect, from a zonally symmetric model of the global energy

balance, incorporating the positive feedback due to the high albedo of snow and sea-ice and seasonally varying insolation, Suarez and Held (1979) get a response qualitatively similar to the geological record over the past 150,000 years when the model is forced with the orbital variations.

Using a modified form of Weer=man's model (1976), ice age continental ice-sheet

growths and decays have also been simulated by Birchfield and Weertman (1978) from insolar.ion anomaly data.

Their model consistently predicts, in addition to significant responses at the

19,000, 23,000 and 41,000 years forcing periods, dominant long-period responses, most commonly at I00,000 and/or 400,000 years.

This, of course, lends support to the hypothesis that

non-llnear response of the climate system to fluctuations of the orbital parameters is responsible for the lonE-period climate fluctuations recorded in deep-sea cores.

Moreover, a general circulation model with a shallow ocean (Mason, 1978) has showed chat, on changing the Earch's orbital p a r m t a r s

from their present-day values to those I0,000 years

ago when cha Earth received 7% m0re solar radiation in June than a~ present, the atmosphere became warmer everywhere, with surface temperatures 6°C higher in the Arctic basin and 4°C higher at 30°N.

From all these results, there seems to be little doubt that the climatic system and the orbital elements are linked by a cause-and-effect relationship, although the'precise linking mechanisms remains unknown.

It is therefore meaningful to search for the percentage of the

climatic variations which can be explained by the astronomical theory alone.

This approach

bypasses an~ solar-terrestrial link and assumes a direct relation (not necessarily linear) between monthly insolation and climate.

4.5.

Insolation Climate Index

To objectively support and/or improve the correlations described in Section 4.3, multivariate analyses have been performed between 6180 deep-sea cores data (Hays et a~.,. 1976) and the zonal monthly mean insolations.

An appropriate homogeneous objective statistical selection

led to monthly insolation for June 85°N and 55°S "(585 and 5 55), December 65°N and 75°S (D65 and D_75) , the intertropical latitudes being represented by March 25°N (M25) and September 15°S (S_15). After different significant

trials

results.

were p e r f o r m e d ( B e r g e r e~ u ~ . , In the f i r s t

1980),

two models gave p a r t i c u l a r l y

o n e , b a s e d upon t h e s p e c t r a l

i d e a , warm and c o o l p e r i o d s *

have been dissociated to test if they are responding differently to the insolation forcing. For the cold period, (chosen as being 15-60,000 YBP), 67% of the total variance can be

*Only the last 135,000 years have been selected to calibrate the model, because of the excellent absolute accuracy of radiometrlc dating over this period.

The N/lankovitch Astronomical Theory of Paleoclimates: A Modern Review

explained using only two principal components.

119

This is significant at the 99% level and the

serial-}? of the residuals is not different from zero at the 90% level.

For the warm periods

(0-15,000 and 60-135,000 YBP), the explained variation is slightly less (still significant), but the serial-R of the residuals is definitely significantly different from zero.

As all

regression coefficients were highly significant, the reconstruction of past climatic variations has been made over the whole time-scale of available geological data (i.e. up to 500,000 YBP).

Then, in contrast to the values mentioned in Section 4.], this model reproduces

as much as 50% of the total climate variations.

The other model is largely based upon the crimatological meaninE and in~lication of the insolation signature concept.

It simulates the dynamic evolution in time of the climatic

response to the insolation forcing, taking into account the memory of the climate system itself.

In fact, persistence has been included and one of the predictors is now the climate

as observed 3000 ?ears before.

Using four principal components to represent the seven input

variables, the model equation becomes for standardized variables: 6180(~) - 0.924 6180(~-3000) + 0.l&8 $85 + 0.110 D65 + 0.004 M25 + 0.032 S_I 5 - 0.036 J-55 - 0.034 D ?5. Surprisingly enough, this relationship explains 87% of the total climatic variation, and the serial-R of the residuals is only 0.29, allowing extrapolation for the next 60,000 ?ears.

The s t a t i s t i c a l geo-eeological

v a l u e o f b o t h ACLIN and INCLIN a e r i e s and t h e i r paleoclimates

authorize

however, will only materialize

the prediction

close agreement with

of the future natural

m o d i f i e d t h e mechanism o f c l i m a t e change and does n o t do so i n t h e f u t u r e . peak w i l l

arrive

climate.

This,

i f m a n ' s i m p a c t on l a n d and t h e a t m o s p h e r e h a s n o t y e t The f i r s t

cold

4000 YAP, and t h e models . f o r e s e e a n improvement p e a k i n g a t a b o u t 15,000 YAP,

f o l l o w e d by a c o l d i n t e r v a l stage four of the last

c e n t e r e d a r o u n d 23,000 YAP.

glacial

cycle,

is indicated

Major glaciation,

c o m p a r a b l e to t h e

a t 60,000 YAP ( F i g . 5 ) .

-I.4-

-123 it,

-33O

~99

6

"6 i @

,< -~

-'2.8-

q,

60

-342

-21

-.4,.2=

I

-300

Fig. 5.

Long-term climatic next 60,000 years. (1976). Full llne Extrapolation into regressive) models: by t h e p e r s i s t e n c e results.

-zoo

-ioo (x I 0 0 0 yr)

o

1oo

v a r i a t i o n s o v e r t h e past,~OO,OOO y e a r s , and p r e d i c t i o n f o r t h e C r o s s e s r e p r e s e n t t h e 6" O d e e p - s e a c o r e s d a t a from Hays e t a l . i s t h e c l i m a t e s i m u l a t e d b y t h e a u t o - r e g r e s s i v e i n s o l a t i o n model. t h e f u t u r e i s b a s e d on r e s u l t s from b o t h ACLIN and INCLIN ( a u t o t h e h i g h f ~ e q u e n c i e s smoothed o u t INCLIN v a l u e s f o r t h e f u t u r e p r e d i c t o r ~ ' " ( t - 3 0 O O ) , h a v e b e e n r e c o n s t r u c t e d from ACLIN

120

A. Berger

All these results also strongly support the key role of some internal processes, like snow and sea-ice (Kukla, 1978), in the mechanism of climate change, and there is no doubt that the final explanation of Pleistocene climates, past as well as future, will have to include interactions of orbital perturbations with non-periodlc terrestrial phenomena.

5. CONCLUSIONS

Assuming paleogeographical configurations and the solar constant being what they were a t the beginning of the Quaternary, recent models, both qualitative and quantitative, conclude that the orbital parameters have m d u l a t e d the climate and will continue to do so assuming no human interferences (Berger, 1980).

As we have very accurate values for orbital elements

and monthly insolations, we must now design both simple models which are able to reproduce the dynamic behaviour of climatic changes and variability, and more sophisticated ones, which allow us to test the validity of the simple models for selected particularly significant dates such as 122,000, 18,000 and 6000 YBP.

Experience already shows that, without any doubt, these

models will have to include seasonal variability of insolation, climate-ocean interactions and albedo-temp era ture-precipita tion feedbacks.

Because of this success of the astronomical theory of paleoclimatas on the Earth, similar studies have been undertaken for Mars (Ward, 1979), where the astronomical effect is even larger due to the planet's orbital properties and where the monthly insolation model will be app lied.

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