Introduction to climate dynamics and climate modelling - Variations of the orbital parameters and of the insolation

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5.4.1 Variations of the orbital parameters and of the insolation

If we ignore the role of the atmosphere, the insolation at a particular time and location at the Earth's surface is a function of the Sun-Earth distance and the cosine of the solar zenith distance (Eq. 2.20). These two variables can be computed from the time of day, the latitude, and the characteristics of the Earth's orbit. In climatology, the Earth's orbit is determined by three orbital parameters (Fig. 5.16 and 5.17): the obliquity ( $ε_{o b l}$ ) measuring the tilt of the ecliptic compared to the celestial equator (Fig. 2.7), the eccentricity (ecc) of the Earth's orbit around the sun and the climatic precession (ecc sin $\tilde{ω}$ ) which is related to the Earth-Sun distance at the summer solstice. In this definition of the climatic precession, $\tilde{ω}$ is the true longitude of the perihelion measured from the moving vernal equinox ( $\tilde{ω}$ = $π$ + PERH on Fig. 2.8).

**Figure 5.16:** Schematic representation of the changes in the eccentricity ecc and of the obliquity $ε_{o b l}$ of the Earth's orbit. Source: Latsis foundation (2001)

Because of the influence of the Sun, the other planets in the solar system and the Moon, the orbital parameters vary with time. In particular, the torque applied to the Earth by the Sun and the Moon because our planet is not a perfect sphere (the distance from the surface to the Earth's centre is larger at the Equator than at the poles) is largely responsible for the variations of the obliquity and plays an important role in the changes in $\tilde{ω}$ . The eccentricity is particularly influenced by the largest planets of the solar system (Jupiter and Saturn), which also have an impact on $\tilde{ω}$ .

The way those parameters have developed over time has been calculated from the equations representing the perturbations of the Earth-Sun system due to the presence of the other celestial bodies and to the fact that the Earth is not a perfect sphere. The solution can then be expressed as the sum of various terms:

\begin{array}{rcl} e c c & = & e c c_{0} + \sum_{i} E_{i} \cos (λ_{i} t + φ_{i}) \\ ε_{o b l} & = & ε_{obl , 0} + \sum_{i} A_{i} \cos (γ_{i} t + ξ_{i}) \\ ecc sin \tilde{ω} & = & \sum_{i} P_{i} \cos (α_{i} t + η_{i}) \end{array}

(5.6)

The values of the independent parameters ecc₀, $ε$ _obl,0, of the amplitudes E_i, A_i, P_i, of the frequencies $λ_{t}$ , $γ$ _i, $α$ _i, and of the phases $φ$ _i, $ξ$ _i, $η$ _i are provided in Berger (1978), updated in Berger and Loutre (1991). The equations (5.6) clearly show that the orbital parameters vary with characteristic periods (Fig. 5.18). The dominant ones for the eccentricity are 413, 95, 123 and 100 ka. For the climatic precession, the dominant periods are 24, 22, and 19 ka and for the obliquity 41 and 54 ka. To completely determine the Earth's orbit, it is also necessary to specify the length of the major axis of the ellipse. However, taking it as a constant is a very good approximation at least for the last 250 million years.

**Figure 5.17:** Because of the climatic precession, the Earth was closest to the Sun during the boreal summer 11 ka ago while it is closest to the Sun during the present boreal winter. Source: Latsis foundation (2001)

The eccentricity of the Earth's orbit (Fig. 5.16) has varied over the last million years between nearly zero, corresponding nearly to a circular orbit, to 0.054 (Fig. 5.18). Using Eq. 2.26, it can be shown that the annual mean energy received by the Earth is inversely proportional to $\sqrt{(1 - e c c^{2})}$ . As expected, this value is independent of the obliquity because of the integration over all latitudes, and is independent of $\tilde{ω}$ because of the integration over a whole year. The annual mean energy received by the Earth is thus at its smallest when Earth's orbit is circular and increases with the eccentricity. However, as the variations in eccentricity are relatively small (Fig. 5.18), there are only minor differences in the annual mean radiations received by the Earth. The maximum relative variation is equal to 0.15% (1.510^-3 = 1 - 1/ $\sqrt{(1 - 0.05 4^{2})}$ ), corresponding to about 0.5 W m^-2 (0.5=1.5 10^-3 x 342 W m^-2).

**Figure 5.18:** Long-term variations in eccentricity, climatic precession and obliquity (in degrees) for the last million years and the next 100 thousand years (zero corresponds to 1950 AD). The minimum value of the climatic precession corresponds to boreal winter (December) solstice at perihelion. Computed from Berger (1978).

The obliquity is responsible for the existence of seasons on Earth. If $ε_{o b l}$ were equal to zero night and day would be 12 hours long everywhere (Eq. 2.22 and 2.25) and if ecc were also equal to zero, each location on Earth would have the same daily mean insolation throughout the year (Eq. 2.22 and 2.26). With a large obliquity, the insolation is much higher in polar regions in summer, while it is zero in winter during the polar night. Over the last million years, the obliquity has varied from 22^o to 24.5^o (Fig. 5.18). This corresponds to maximum changes in daily mean insolation at the poles of up to 50 W m^-2 (Fig. 5.19). Obliquity also has an influence on the annual mean insolation, increasing it by a few W m^-2 at high latitudes and decreasing it (but to a lesser extent) at the Equator.

**Figure 5.19:** Changes in the seasonal contrast of insolation in W m^-2 caused by (top) an increase in the obliquity from 22.0^o to 24.5^o with *ecc*=0.016724, PERH=102.04, i.e. the present-day values, and (bottom) following an increase in the climatic precession from its minimum value (boreal winter at perihelion) to its maximum value (boreal summer at perihelion) with *ecc*=0.016724, $ε_{o b l}$ =23,446^o, i.e. the present-day values. Contour interval is 10 W m^-2. The brown areas correspond to zone with a zero insolation. Time of the year is measured in term of true longitude $λ_{t}$ . It is assumed that $λ_{t}$ =-80^o corresponds to the 1^st of January and one month corresponds to 30^o in true longitude.

Finally, the position of the seasons relative to the perihelion (i.e., the precession) also has an influence on insolation. If Earth is closer to the Sun during the boreal summer and further away during the boreal winter, the summer in the northern hemisphere will be particularly warm and the winter particularly cold. On the other hand, if the Earth is closer to the Sun during boreal winter, the seasonal contrast will be smaller in the northern hemisphere. This effect is particularly marked if the eccentricity is large. Actually, if the eccentricity is nearly zero, the distance between the Earth and the Sun is nearly constant, implying no impact of the changes in the position of the seasons relative to the perihelion. The climatic precession varies roughly between -0.05 and 0.05. This induces changes in insolation that can be greater than 20 W m^-2 at all the latitudes (Fig. 5.19). As a consequence, the climatic precession effect dominates the variations in insolation at low and mid latitudes.