Theor. Appl. Climatol. 53,253-255 (1996)

Theoretical and Applied ClimatoFogy © Springer-Verlag 1996 Printed in Austria

Comments to "Insolation in terms of Earth's orbital parameters" by D. P. Rubincam Theor. Appl. Climatol. 48, 195-202, 1994 A. Berger Received March 3, 1994 Revised November 30, 1994

Solar irradiation received on the Earth - generically called i n s o l a t i o n - is a key factor in climate and climatic variations research. This is because solar energy is the primary source of energy driving the climate system and also because it can be easily and accurately computed. The calculation of the instantaneous energy received from the Sun at a specific location (of latitude ~band longitude 2) on the Earth is a geometrical problem which is usually taught in astronomical courses (Milankovitch, 1920, 1941). Its integration over a particular time interval (hours, days, months or a year) and over a given surface of the Earth (a zonal band, an hemisphere, a sector or the whole Earth) is a mathematical problem which is easily solved by numerical procedures but which is not as straightforward if an analytical formula has to be found (e.g., Berger, 1975; Loutre, 1993). The time variations of this insolation depends upon the energy output from the Sun (an astrophysical problem) and upon the astronomical parameters characterizing the orbit of the Earth around the Sun and its axis of rotation (a problem which is relevant to celestial mechanics and geophysics). Those astronomical factors are the eccentricity of the Earth's orbit, the longitude of the perihelion and the obliquity (see for example Milankovitch, 1941). In this paper, we will note these parameters e, co and O respectively as done in Rubincam (1994) although a more conventional notation is e, (7) and e (e.g., Berger, 1978).

For trying to understand the behavior of the insolation as a function of these astronomical factors, the analytical formula obtained from geometrical considerations is sometimes developed using spherical .harmonics. This is what has been done recently by Rubincam (1994). The purpose of our comments is related to his conclusion that "the most easily attained insights- gained from his equation for solar insolation - are that neither the main pacemaker of the ice ages, e, nor Milankovitch's precession index e sin co appear as terms in the equation. Obliquity does appear." The following comments will demonstrate that this is a misinterpretation of the equation obtained by Rubincam (1994), in particular his Eq. (8) page 200, and therefore that his conclusion is not correct. In order to be able to refer easily to the material already published by the authors and their collaborators, let us first remind the fundamental equation for the instantaneous insolation, F s (the irradiance expressed in W m - 2). Using the notation by Rubincam (1994), Eq. (7) of Berger et al. (1993) can be written as Eq. (8) of Rubincam (1994): Fs = F ° H

,o

costs

(1)

", ]"s /

where F s° is the solar irradiance for a reference distance r 0 from the Sun to the Earth, Os is the solar zenith angle and H takes the value 0 or 1 whether the Sun is below or above the horizon.

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A. Berger

For the sake of comparison, F S, F s, ° rs and Os are noted W, S, r and z in Berger et al. (1993) and r o is taken as being the semi-major axis, a, of the Earth's orbit. In this case (1) can be written more explicitely

F~ = F ° H (1 + e cos(2~ - o)))2

(1

-- e2) 2

x (sin ~bsin a + cos 4~cos a cos ~/)

(2)

where q is the hour angle of the Sun, a its declination and 2s its ecliptical longitude (~/and 2, are noted H and 2 in Berger et al., 1993) 2~ and 6 are related by: sin a = sin O sin 2~. Instead of considering one particular time of the day given by ~/, (2) can be integrated from sunrise to sunset. The mean irradiance (over 24 hours), ff~, is then given by 1. where there is a sunrise and a sunset p = s

(~/o sin q~sin 6 + cos q~cos 6 sin r/o) 7~

(3) where r/o is the solar hour angle at sunset and r and 6 are assumed to be constant over the day. 2. where there is a long polar night (i.e., no sunrise) F~ = 0

(4)

3. where there is a long polar day (i.e., no sunset) ff~= F°s \(-a)2 r / sin ~bsin 6

(5)

F r o m the analysis of (3), (4) and (5), analytically and numerically in the spectral domain, Berger and Pestiaux (1984) and Berger et al. (1993) concluded that the climatic precession parameter (esinco) dominates for all days and latitudes, although the obliquity is more d o m i n a n t in high than low latitudes. The only exceptions for which obliquity dominates precession is for latitudes and days close to the polar night. Clearly this is in total opposition to the conclusion of Rubincam (1994) stated in particular pages 195, 200 and 201, where we find: "So, e sinco cannot appear in the insolation, and therefore Milankovitch's (1941) precessional index, the change A(e sin co), cannot be significant in terms of the insolation." As the formula given here are

equivalent to the zonal insolation of Rubincam (his page 200), why is this so? The answer is quite simple. In his formula (8), Rubincam (1994) has terms in sin O sin co and in 2e sin O sin co. These terms are providing implicitely a very strong spectral power to precession because the time variation of sin O is dominated by sin O* where O* is a constant (e.g. Sharaf and Budnikova, 1967; Berger, 1978). F r o m Eq. (1) in Berger (1978), one has indeed:

0 = O* + ,r, Ai cos(flt + 6i) As for the Quaternary O is varying roughly between 22 ° and 25 °, we have 0.375 ~< sin O ~< 0.423 and the variations around the constant value, sin O* --- 0.396, are of the order of _+ 6~o only. In addition of computing spectral analyses of time series of ffs values over a given interval of time (150,000 years for example, as illustrated in Berger et al., 1993), the dependence of ffs in terms ofe sin co can be shown in one specific case at least. The zonal insolation available at latitude ~b, at the vernal equinox, is not depending on obliquity at all, but only u p o n precession. F r o m a geometrical point of view, this is because at that time, the Earth's axis of rotation is lying in a plane perpendicular to the sun rays. At this date, the daily irradiation, to an accuracy comparable to the one of formula (8) of Rubincam (1994), is indeed given by:

ff s = F° ( s rc

e2

5e 2

1 + 2e cos co + - - cos 2co + - 2 2

+ 4e 3 cos co) cos ~b which is only a function of precession and of e 2. Surely, our formula (2) shows also (as recalled by Rubincam page 201) that at the perihelion position (i.e., 2 s = c0), ffs is not depending u p o n precession at all, but is only a function of eccentricity and obliquity. The distance factor, (a/r) 2, in this case gives rise to a factor which is equal to 1/(1 - e) 2, reinforcing considerably the influence of the eccentricity, as this factor is roughly equal to (1 + 2e + 3e2). For the more elongated orbit (e having a m a x i m u m value of roughly 0.07), this factor is contributing for an insolation 15}/o larger than for a circular orbit (e = 0). However, we have

Comments to "Insolation in terms of Earth's orbital parameters"

to recognize that the perihelion position drifts continuously through all the seasons according to the long-term variation of its longitude (precession). The physical mechanisms through which the climate system would respond to such a forcing are therefore very difficult to conceive (Berger, 1989, p. 50). Finally, let us stress that the mathematical derivation of the equations given in Rubincam has not been checked by the authors of this commentary note. The mathematical treatment described all the way through the paper by Rubincam (1994) is most probably correct. It is his interpretation of how the insolation varies on the time scales of the Milankovitch cycles - one of the purposes of the author to derive his equation for the solar insolation, as he says in his paper - that we claim not being correct.

References

Berger, A., 1975: D6termination de l'irradiation solaire par les inthgrales elliptiques. Annales Socidt~ scientifique de Bruxelles, 89(1), 69-91. Berger, A., 1978: Long-term variations of daily insolation and Quaternary climatic changes. J. Atmos. Sci., 35(12), 23622367. Berger, A., 1989: The spectral characteristics of pre-Quaternary climatic records, an example of the relationship between the astronomical theory and Geo-Sciences. In: Berger, A., Schneider, S., Duplessy, J. C1. (eds.) Climate and

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Geo-Sciences, a Challenge for Science and Society in the 21st Century. 47-76. (NATO ASI Series C: Mathematical and Physical Sciences, vol. 285) Kluwer Academic Publishers: Dordrecht, Holland. Berger, A., Pestiaux, P., 1984: Accuracy and stability of the Quaternary terrestrial insolation. In: Berger, A. Imbrie, J., Hays, J., Kukla, G. Saltzman, B. (eds.) Milankovitch and Climate. Reidel: Dordrecht, Holland, 83-112. Berger, A., Loutre, M. F., Tricot, C., 1993: Insolation and Earth's orbital periods. J. Geophys. Res., 98, D6, 10,341 10,362. Loutre, M. F., 1993: Param6tres orbitaux et cycles diurnes et saisonniers de l'insolation. ThSse de doctorat, Facult6 des Sciences, Universit6 catholique de Louvain, Louvain-laNeuve, 140 pp. Milankovitch, M., 1920: Th6orie mathhmatique des phbnomhnes thermiques produits par la radiation solaire, 339 pp., Acad: Yougoslave des Sci. et des Arts de Zagreb, Gauthier-Villars, Paris. Milankovitch, M., 1941: Kanon der Erdbestrahlung und seine Anwendung aufdas Eiszeitenproblem, 633 pp., Ed. Sp. Acad. Royale Serbe, Belgrade. (English translation Canon of Insolation and Ice Age Problem, Israel program for Scientific Translation; published for the U.S. Department of Commerce and the National Science Foundation). Rubincam, D. P., 1994: Insolation in terms of Earth's orbital parameters. Theor. Appl. Climatol., 48, 195-202. Sharaf, S. G., Budnikova, N. A. 1967: Secular variations of elements of the earth's orbit which influence the climate of the geological past (in Russian). Tr. Inst. Teor. Astron., 11(4), 231-261. Author's address: Dr. A. Berger, Institut d'Astronomie et de Geophysique. Georges Lemaltre (Unit6 Astr), Universit6 Catholique de Louvain Chemin du Cyclotron, 2, B-1348 Louvain-la-Neuve, Belgium.