To resolve the temporal variability of insolation requires a sampling interval of less than a day,2 and for most paleoclimate contexts, extending over thousands of years
— a prohibitively large data-set. Ninety such timeseries are required if one seeks two degree latitudinal resolution. Thus, for many applications, simplified versions of the insolation are instead used, often which aim to reduce the variability to a single timeseries requiring only a low sampling resolution. This over simplification inevitably enhances some modes of variability while diminishing or suppressing others.
In this section some of the most common insolation simplifications are reviewed and compared with one another. In the next section a more complete description of the insolation is developed.
Spatial simplifications of insolation forcing usually take the form of choosing a particular latitude [e.g. Milankovitch, 1941], the difference between two latitudes [e.g. Raymo and Nisancioglu, 2003], or averaging over a hemisphere or the globe [e.g.
Berger, 1978]. Common temporal simplification of the full forcing function are to
2Technically, because of the day-night clipping of insolation, a much shorter sampling interval is required to perfectly reconstruct the signal, but in practice hourly samples provide a good approxi-mation.
M A M J J A S O N D J F M
Figure 2-10: The root-mean-square insolation variability due to changes in(a) obliq-uity and (b)precession, shown as a function of latitude and day of year running from March 22nd (vernal equinox) to March 21st. Because the root-mean-square variabil-ity is only calculated on a single day of the year, it is rectified, and thus has precession period variability. Contour lines are labeled in W/m2.
pick a particular solar-longitude, day of the year, or some interval of time or solar longitude [e.g. Vernekar, 1972]. Depending on the location and day selected, different modes of secular variability are more pronounced. Also note that the solar longitude and day of the year do not have a unique relationship [e.g. Figure 2-1; Vernekar, 1972; Berger et al., 1993]. Joussaume and Braconnot [1997] show that paleoclimate simulations which do not take into account changes in the seasonal cycle can have biases of the same order as the simulated climate change.
Figure 2-10 shows the root-mean-square (rms) insolation variability due to obliq-uity and precessional variations, contoured as a function of latitude and day of the year. Insolation calculations are made using the orbital solution ofBerger and Loutre [1992] and a program provided by J. Levine [personnel communication] which calcu-lates insolation based on Earth’s position and orientation relative to the sun. The program has been modified to run more efficiently and to calculate insolation either according to day of the year or to solar longitude. At the equinoxes, currently occur-ring on March 20th and September 22nd, earth’s tilted spin axis is perpendicular to the direction of the sun, and obliquity has no effect on insolation. Insolation varia-tions due to obliquity are the most pronounced during summer and have the opposite
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Figure 2-11: ((a)Astronomical and (b)caloric half-year insolation averaged for lat-itudes ≥ 60◦ where averages are area weighted. The corresponding periodograms are plotted at right with vertical dashed lines at frequencies of 1/41KY (obliquity) and 1/21KY (precession). Note that calculating insolation only over half the year is a rectification which generates the precession period variability. In addition, the astronomical half-year insolation is averaged over a period which itself depends on precession. The caloric half-year and calendar half-year insolation (not shown) are both averaged over fixed periods and are very similar with a squared-cross-correlation greater than 0.99. (c) Caloric half-year insolation averaged between zero and 30◦N along with its periodogram (right). At low-latitudes, the astronomical and caloric half-years are both dominated by precession and are very similar to one another.
sign during winter. At high-latitudes, increases in obliquity cause small decreases in winter insolation and larger increases in summer insolation, so that the net effect is to increase the annual insolation. At lower-latitudes, the net winter decrease in insolation due to obliquity has a greater magnitude than the summer increase, with the cross-over in seasonal influence occurring at 43◦ North and South. This cross-over can be derived from the insolation equations of Rubincam [1994] by omitting all the annual and higher frequency terms and solving for the zeros. Therefore, depending on the selected date and latitude, the variability due to obliquity or precession can vary substantially.
Rather than calculating insolation for a given day or solar longitude, it is possible
to average insolation over some portion of the year. For the case of half-year insolation, several possibilities exist: astronomical, calendar, and caloric half-year insolation (see Figure 2-11). (1) Astronomical half-year insolation is obtained by averaging between solar-longitudes, λ = 0 and λ = π [e.g. Vernekar, 1972]. Although the seasons are strictly defined, the duration of the averaging period changes, according to the degree of eccentricity and the angle of perigee. The earliest and latest dates corresponding to the autumnal equinox (λ=π) are separated by a month.
(2) Calendar half-year insolation averages between two selected dates — usually vernal equinox and somewhere near the time variable autumnal equinox. The du-ration of the calendar half-year is fixed, but because the date of autumnal equinox is time-variable, it does not ensure a true winter or summer season. This prompted Milankovitch [1941] to introduce: (3) the caloric half-year which maintains equal du-rations between the summer and winter half-years, while maximizing the insolation contrast [e.g. Milankovitch, 1941]. Vernekar [1972] and Berger [1978] give good discussions of both astronomical and caloric half-year insolation. The astronomical half-year insolation has more precession variability than either the calendar or caloric half-year insolations owing to precession’s influence on both the intensity of the inso-lation and period over which the insoinso-lation is averaged. It is also possible to average over some latitude band (see Figure 2-11); when the average is confined to only high or low latitudes, the variance and frequency structure of the signal changes only slightly.
There exists an endless number of possibilities regarding the temporal and spa-tial averaging of insolation. Various choices can be motivated by physical concepts, but in the absence of a general theory for how changes in radiation at the top of the atmosphere affect climate, many choices remain plausible but largely arbitrary.
Furthermore, no low-frequency timeseries of insolation can represent how the secular changes in insolation affect both the seasonal and low-frequency changes in insolation.
As the annual cycle is extremely powerful3, this constitutes a major short-coming of traditional simplification of the insolation variability. In Chapter 1, it was shown how neglecting high-frequency variability could lead to incorrect power-law estimates; and in Section 2.3 it was shown how the proxy response to the annual cycle could be misin-terpreted as low-frequency climate variability. For these reasons, a full representation of the seasonal and secular insolation variability is important for understanding the
3Consider that the annual range in Arctic temperatures is on the order of 50◦C, roughly double the difference between mean temperatures during the Holocene and the Last Glacial Maximum calculated byDahl-Jensen et al. [1998] using the GRIP borehole.
relationship between climate and orbital forcing.