Intertropical Latitudes and Precessional and Half-Precessional Cycles

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Abstract
A. McIntyre and B. Molfino, in their report about the forcing of Atlantic equatorial and subpolar millennial cycles by precession, suggest that climatic changes in high polar latitudes (which are related to Heinrich iceberg surges) may be caused by events that occur in low latitudes ([1][1]). This suggestion is based partly on a quasi cycle of about 8400 calendar years that they found in the relative abundance of a tropical marine algae and that they relate to precession. Our comment is about their attempt to give this period an astronomical origin. ([1][1], p. 1869). 1) In the intertropical zone, the sun passes directly overhead twice in a year at each latitude, but this does not imply that “[o]ver one precessional cycle, this produces two intervals during which perihelion is coincident with the solstice in Northern Hemisphere summer,” as stated by McIntyre and Molfino ([1][1], p. 1869). By definition of the precessional cycle, perihelion can coincide only once with the Northern Hemisphere summer solstice during one cycle. What produces half a precession cycle in the tropics can only be explained if we accept the proposition that, in the tropics, the climate is responding principally to the largest maximum of insolation, independently of the date it occurs during the year. At the equator, for example, the sun passes overhead at both equinoxes. The evolution of the daily insolation at the equator at the spring and autumn equinoxes can be graphed (Fig. [1][2]). This insolation is given by [see formula 30 in ([2][3])]![Formula][4](1)where W is the insolation; S is the absolute solar constant, estimated at a distance equal to the semimajor axis of the Earth orbit around the sun; e the eccentricity; and ω̃ the so-called longitude of the perihelion (ω̃ = 0 when the spring equinox occurs at the perihelion;ω̃ is currently equal to 282°). ![Figure 1][5] Figure 1 Long-term variation of the daily insolation at the spring and autumn equinoxes at the equator, 100 kyr BP to 100 kyr AP ([3][6]). These insolations do not depend at all on obliquity. Their spectrum is dominated by precession [about 23- and 19-kyr (thousand-year) periods], but displays also, with much less power, half-precessional periods (11.5 and 9.5 kyr), eccentricity periods, and combination tones. To a good approximation, equation (1) can be written![Formula][7] ![Formula][8](2)Equation [2][9] shows also that, with an excellent approximation,![Formula][10](3) ![Formula][11](4)and therefore that the insolations at spring and autumn equinoxes are out of phase by half a precession cycle. Selecting for each date of the past, the largest of the two values, leads to a curve (Fig. [2][12]) that is dominated by half-precessional cycles. ![Figure 2][5] Figure 2 Long-term variation, 100 kyr BP to 100 kyr AP, of the largest of the two values of the daily insolation at the spring and autumn equinoxes at the equator ([3][6]); see also ([8][13]). This result conceptually remains valid for intertropical latitudes φ : −ɛ ≤ φ ≤ ɛ, although it is then slightly more complicated because of the additional role played by obliquity outside the equator. 2) We agree that the primary components in the tropics are the precessional periods. But this holds also for high polar latitudes (except for those very close to where the polar night occurs [see figures 10B and 14B in ([2][3])]. Moreover, the periods of the two largest amplitude terms in the trigonometrical expansion of climatic precession are 23,716 and 22,428 years ([3][6]), which average (weighted or not) is about 23,100, not 22,000 years, as considered by McIntyre and Molfino ([1][1]) with a first harmonic of 11,550 years, not 11,000 years. 22,000 years is much more related to the average period of the precessional parameter itself than to its spectral components. This period is not stable in time and, over the last 60 kyr, we observe its progressive shortening (Fig. [3][14]) : the minima of the climatic precession occur at −61, −33, −12, and +9 kyr, respectively, which leads to periods that are 28, 21, and 21 kyr long, respectively; if we consider the interdistances between maxima (which occur at −47, −22, −1, and +18 kyr), we arrive at 25, 21, and 19 kyr. ![Figure 3][5] Figure 3 Long-term variation of eccentricity, climatic precession, and obliquity between 100 kyr BP and 100 kyr AP. See the damping and shortening of the cycle of eccentricity and precession between now and 50 kyr AP ([3][6], [9][15]). One reason for this result is the rather particular behaviour of eccentricity as quoted by the authors [see also ([4][16])]. The minima of e occur at −158, −45, and +27 kyr, leading to a length of the period equal to 113 and 72 kyr; the maxima occur at −115, −14, and 87 kyr, leading to two periods of 101 kyr (Fig.[3][14]). Asymmetry, damping, and associated shortening of the eccentricity cycle are therefore important because they influence significantly the length of the precessional cycle. If 22 kyr is important in the finding of McIntyre and Molfino ([1][1]), we would suggest that it corresponds to an average value of the precessional cycle over the last 45 kyr. This result is not without having consequences on the spectral components of precession themselves. The eccentricity cycle and all the spectral components of eccentricity are obtained as combinations of the precession frequencies. For example, we see ([5][17], p. 638) that the second period of e (94,945 years) comes from the periods of terms 3 and 1 of the expansion of precession:![Formula][18](5)If we consider the shortening of the eccentricity and precession cycles over the period between roughly 100 kyr BP and 30 kyr AP, we may give an astronomical interpretation to the additive combination tone calculated by McIntyre and Molfino. By calculating (1/72 + 1/22 = 1/16.85) they tentatively compute...

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