Is the result of Earths orbit around the Sun and the degree of the Earths axis in the ecliptic plane?

Axial Angular Momentum in Regions

The dominant relative angular momentum about Earth's axis depends on the strength of zonal winds, which tend to be persistent features of the atmospheric circulation. In Figure 1 we present the latitude–pressure distribution of the long-term zonal mean zonal wind based on fields from the reanalysis dataset from the U.S. National Centers for Environmental Prediction and National Center for Atmospheric Research (NCEP–NCAR); we used a 30-year period at the end of the twentieth century to form a climatology of the mean zonal wind. A similar signature exists in the two hemispheres. Mean easterly winds are found in the tropical regions, with a broader latitudinal extent at the surface than higher in the troposphere. The winds are westerly over most of the extratropics. Increases in the strength of these winds with height in the atmosphere lead to very strong westerlies in the upper troposphere, at levels near 200 hPa; above this jet level, they tend to decrease again. Regional maps of such jets at this level reveal that the strongest values are located over the eastern North American and Asian continents. The strong winds at these regions contribute heavily to the relative component of the axial angular momentum of the atmosphere.

Angular momentum can be computed in zonal belts from values of the zonal wind, so that a profile of angular momentum (Figure 2) reveals the general distribution with latitude. The seasonality of the angular momentum can be noted as the substantial difference between the December–January–February periods and the June–July–August periods. It is clear that middle latitude belts have their largest values during their winter, in both the Southern and Northern Hemispheres, but the annual cycle is larger in the northern than in the southern hemispheric belts. Most of the atmosphere has westerly relative angular momentum, indicating that in these regions the atmosphere superrotates with respect to the underlying planet.

Obliquity

The obliquity (tilt) ε of Earth's axis with respect to the orbital plane (see Figure 1) is defined by the angle between Earth's spin vector s and that of the orbital plane n, and can be computed as cos ε = n 

s, using unit vectors. As the inclination and orientation of the orbital plane vary, the obliquity is not constant, but oscillates due to the interference of the precession frequency p and the orbital elements si. As shown in Table 3, if the variation in obliquity is approximated by quasiperiodic terms, the result is a strong oscillation with a period of approximately 41 ky, with additional periods around 54 and 29 ky. The ∼41-ky period arises from the simultaneous variation in Earth's orbital inclination, given by s3, and the precession of Earth's spin direction, expressed by p. Table 3 also shows that the obliquity signal contains contributions from the gi as well as the si fundamental frequencies, due to their combined effect on the change of the orbital plane normal. The present day obliquity of approximately 23.45° has varied between ∼22.25° and ∼24.5° during the past 1 million years. The main climatic effect of variations in Earth's obliquity is in control of the seasonal contrast. The total annual energy received on Earth is not affected, but the obliquity controls the distribution of heat as a function of latitude, and is strongest at high latitudes.

Table 3. Six leading terms for Earth's obliquitya

It is important to note that all of the obliquity frequency components contain the precession constant p. Due to tidal dissipation, the frequency of the precession constant p has been higher in the past, a fact that can be shown from geological observations, such as ancient growth rings in corals, and from tidal laminations. Figure 5 illustrates the variation in obliquity over 1.2 million years. The oscillation is dominated by a ∼41-ky period cycle, and a variation in amplitude is also observed. This variation is due to beats arising from the presence of additional ∼29- and ∼54-ky periods, which are just visible in the frequency analysis shown on the right-hand side of Figure 5.

3.01.3.3.1 Declination angle, δ

As shown in Figure 3, the earth axis of rotation (the polar axis) is inclined at an angle of 23.45° from the ecliptic axis, which is normal to the ecliptic plane. The ecliptic plane is the plane of orbit of earth around the sun. The solar declination angle is the angular distance of the sun’s rays north (or south) of the equator, north declination designated as positive. As shown in Figure 5, it is the angle between the sun–earth center line and the projection of this line on the equatorial plane. Declinations north of the equator (summer in the Northern hemisphere) is positive and those south are negative. Figure 6 shows the declination angle during the equinoxes and the solstices. As can be seen, the declination angle ranges from 0° at the spring equinox, to +23.45° at the summer solstice, to 0° at the fall equinox, to −23.45° at the winter solstice.

The variation of the solar declination angle throughout the year is shown in Figure 7. The declination angle δ, in degrees, for any day of the year (N) can be calculated approximately by the equation (ASHRAE, 2007):

[5]δ =23.45sin[360365(284+N)]

Declination can also be given in radians by the Spencer formula [15]:

[6]δ=0.006918−0.399912cos(Γ) +0.070257sin(Γ)−0.006758cos(2 Γ)+0.000907sin(2Γ)−0.002697cos(3Γ)+0.00148sin(3Γ)

where Γ is called the day angle given by (in radians):

[7]Γ=2π(N−1)365

The solar declination during any given day can be considered constant in engineering calculations [16, 17].

As shown in Figure 6, the tropics of Cancer (23.45°N) and Capricorn (23.45°S) are the latitudes where the sun is overhead during summer and winter solstice, respectively. Another two latitudes of interest are the Arctic (66.5°N) and Antarctic (66.5° S) Circles. As shown in Figure 6, at winter solstice all points north of the Arctic Circle are in complete darkness, whereas all points south of the Antarctic Circle receive continuous sunlight. The opposite happens for the summer solstice. During spring and fall equinoxes, the North and South Poles are equidistant from the sun and daytime is equal to nighttime, which are both equal to 12 h.

5.1.5 The Nauruz [Nowruz; The Iranian New Year] and the Bull

According to an ancient Iranian legend, the Earth's axis turns on one horn of a giant bull. Once a year, on the vernal equinox, the bull tosses its burden from one horn to the other so deftly that the shift can be observed only by watching the delicate movements of a highly sensitive egg on a polished, slippery surface. The traditional Iranian Nauruz ritual is setting a (table) cloth [“sofreh”] with seven iconic dishes corresponding to the seven creations and the seven holy immortals protecting them. A bowl of clear water [representing the sky], containing an egg or an orange [representing the planet Earth] and either a leaf of a rose or a green leaf, and a mirror [representing sincerity] are placed on a table. A plain hard-boiled egg is placed on the middle of the mirror. It is believed that at the exact moment of the New Year, when the Sun enters into the new sign of the zodiac (the spring equinox; first day of the Iranian new year), the bull tosses the Earth from one horn to the other horn. Consequently, the egg rolls on the mirror, the orange flips over in the bowl of water, and the floating leaf moves (Donaldson, 1933; Massé, 1938; Messner, 1954; iranicaonline.org/nowruz).

Satellite

Satellite orbits have fundamental characteristics that principally vary with their radius (altitude) and orientation with respect to the Earth's axis of rotation. The geostationary orbit has an altitude of 36,000 km orthogonal to rotational axis, which implies that satellites in this orbit have the same angular speed as the Earth and thus appear stationary to one portion of the Earth's surface. This orbit is seldomly used in geology, but is important for communication and weather satellites for obvious reasons. The large majority of imaging sensors are on sun-synchronous, near-polar orbits. This orbit allows the satellite to fly over the same area at the same time of the day and provides a nearly global coverage of the Earth over a period of typically 1–2 weeks. Sun-synchronicity is particularly important as it allows consistent illumination conditions in a given area and thus facilitates mosaicking and temporal comparisons. The disadvantage is that half the orbits are in the shadowed side of the Earth and, therefore, are not suitable for most passive sensor applications. Orbits used for the global navigation satellite system (GNSS) are of intermediate distance from the Earth, typically 20,000 km, in order to maximize coverage and minimize the number of satellites in the constellation. Finally, due to their altitudes typically between 400 and 1000 km, sensors on orbital platforms will have characteristics that trade-off between swath width and spatial, spectral and temporal resolutions. While scientific missions such as LANDSAT or ENVISAT were historically predominant, hybrid (e.g., RapidEye, Terrasar-X) or purely commercial (e.g., worldview, geo-eye) solutions are becoming increasingly popular. The rising demand for high-resolutions data has led to on-demand acquisitions over small areas, which limits time series analysis and continuous archives.

I.E.1 Effective Gravity

An object at rest in a coordinate system rotating with the earth is subject to a centrifugal force directed outward from the earth's axis and a gravitational force directed toward the earth's center. An observer or an instrument on the earth cannot distinguish between these two forces. Therefore they are combined into a resultant force

(25)g=g0−Ω×Ω×r

known as the effective gravity or simply gravity. The local vertical is defined to be in the direction of this force. Note that it points toward the center of the earth only at the poles and equator.

Effective gravity

An object at rest in a coordinate system rotating with the earth is subject to a centrifugal force directed outward from the earth's axis and a gravitational force directed toward the earth's center. An observer or an instrument on the earth cannot distinguish between these two forces. Therefore they are combined into a resultant force

[25]g=g0−Ω×Ω×r

known as the effective gravity or simply gravity. The local vertical is defined to be in the direction of this force. Note that it points directly toward the center of the earth only at the poles and equator.

3.6 Coriolis

The Earth rotates around its own axis from west to east. Therefore, a reference frame attached to a fixed position on the Earth rotates around the Earth axis. This leads to a complication when we try to apply Newton’s law of motion on a coordinate system that is attached to the Earth, specially when studying large-scale ocean circulation and tidal dynamics. An object (or here, our reference frame), which rotates around an axis, has acceleration (i.e. centripetal acceleration), because its velocity is changing with time and Newton’s law of motion is not valid in a frame, which has an acceleration. Whilst the speed (scaler) of an object rotating around an axis may be constant, its velocity (vector) changes due to the change in direction. To further clarify this concept, consider an object on the surface of the Earth which rotates around the Earth with an angular velocity of Ω (Fig. 3.10). The speed of the object will be u = REΩ, where RE is the radius of the Earth. The velocity of the object with respect to a nonrotating frame at the centre of the Earth is given by

(3.9)u→=−usinθî+ ucosθĵ=REΩ[−sinθî+cosθĵ]

Therefore, we can calculate the acceleration by taking the derivative of velocity as follows

(3.10)a→=du →dt=REΩ−cosθdθdtî−sinθdθdtĵ=−REΩdθdt[cosθî+sinθĵ]

Because Ω=dθdt, then

(3.11)a→=−REΩ2[cosθî+sinθĵ]

Eq. (3.11) describes a vector, which is orientated towards the centre of the earth, with a magnitude of REΩ2 = u2/RE, which is called the centripetal acceleration.

In order to apply the equations of motion in a rotating frame attached to the Earth, we need to find the relationship between rotating and nonrotating frames. In general, an object can rotate around three axes in space; therefore, to deal with a rotational coordinate system in a more general way, we need to formulate a relationship between inertial and rotational frames of reference in 3D. Consider a frame of reference at the centre of the Earth (inertial), and a rotational frame of reference at a point in the ocean (rotates around the Earth’s axis). The rotating frame may be described by three unit vectors, îr, ĵr, and k^r. These unit vectors are rotating with angular velocity Ω→ around the inertial frame of reference. If we represent the position of an object in 3D using the position vector R, the rotational speed will be RΩ, or angular velocity will be Ω→×R. Based on the theory of relative motion in dynamics, for any vector variable A, we can write

(3.12)dAdtiner=dAdtrot+Ω→×A

Consequently, the relationship between velocities in the inertial and rotating frame can be written as

(3.13)dRdtiner=dRd trot+Ω→×R⇒uiner= urot+Ω→×R

Similarly, if we apply Eq. (3.12) for acceleration (as a vector), it leads to

(3.14) duinerdtiner=duinerdtrot+Ω→×uiner

Because, u iner=urot+Ω→×R, therefore,

(3.15)duinerdtiner=d(urot+ Ω→×R)dtrot+Ω→ ×(urot+Ω→×R)

(3.16) =durotdtrot+2Ω→×urot+ Ω→×(Ω→×R)+DΩ→dt×R

As can be seen, the acceleration, which is observed in a rotational frame (i.e. durotdt), is different from the acceleration, which is observed in an inertial frame (i.e. duiner dt). The difference between these accelerations includes three terms: Coriolis acceleration (−2Ω→×urot ), the centripetal acceleration −Ω→×(Ω→×r), and, Euler acceleration, the acceleration due to change in the angular velocity (−DΩ→dt×R). These terms appear when we write the equation of motion in a rotational coordinate system. Therefore, we refer to them as apparent or fictitious accelerations or forces. Assuming that the angular velocity of the Earth is constant results in dΩ→dt=0. The centripetal acceleration at the equator is about 0.3% of the Earth attraction, and in measurements it is difficult to differentiate between the Earth attraction and centripetal acceleration. The resultant force of Earth attraction and centrifugal force is called gravity, which acts normal to the surface of the Earth.

Now, consider a rotating frame attached to the Earth, in which x and y axes correspond to the west-east and the south-north directions, respectively. The z axis is oriented towards the centre of the Earth. An object moving northwards in the northern hemisphere is considered; using the cross product, the magnitude of Coriolis acceleration will be, 2Ωsinϕv, where ϕ is the latitude (or the angle between velocity and rotation vectors), and the direction of the Coriolis acceleration will be towards the east, or to the right of moving object (i.e. perpendicular to the plane containing rotation and velocity vectors). It can be seen that the magnitude of Coriolis is zero for an object moving northwards at the equator (ϕ = 0).

Let us assume that an object is moving from the west towards the east in the northern hemisphere; the magnitude of Coriolis acceleration (using cross product) will be 2Ωu, as the rotation vector and velocity vector are perpendicular. For this scenario, the Coriolis acceleration has two components: one component towards the south, or negative y direction. With a magnitude of 2Ωusinϕ, and the other component away from the centre of the Earth (z axis), with a magnitude of 2Ωucos ϕ. Therefore, the Coriolis effect turns objects rightward in the northern hemisphere. We can similarly show that Coriolis turns objects leftward in the southern hemisphere.

As we know, the Earth rotates 2π in 1 day. At the same time, it also rotates a little (≈1/365 × 360degrees) around the Sun. A sidereal day (86,164 s) is the exact time that it takes for the Earth to make one rotation, and is slightly shorter than a solar day (86,400 s). Therefore, the angular velocity of the Earth is Ω=2π86,164=7.29×10−5 per s. Note that the Coriolis parameter or frequency, f, is defined as 2Ωsin(ϕ).

Geology of Reservoirs

Humans have stored so much water in reservoirs that it has been suggested to have subtly altered the planet's rotation. Water impoundment is thought to have shortened the length of the day and shifted the Earth's axis by tiny amounts. No other human activity has been big enough to cause any appreciable alteration in these global phenomena. Large reservoirs can also cause earthquakes because of their addition of heavy mass. The largest reservoir-induced earthquake occurred in 1967 in India and had a magnitude of 6.3 or 6.8 on the Richter scale. Reservoirs are also recognized for their ability to collect sediments. All rivers erode their beds and carry sediments downstream. Large reservoirs trap most of these sediments because currents slow down and drop their sediment loads when entering the reservoirs. Relatively sediments-free water is then released from the reservoir, in turn eroding new sediments from the channel below the dam and carrying them downstream. Because no sediments are deposited where this erosion is happening below the dam, the erosive capacity of the water may lower the riverbed by several meters until it is armored by stones and boulders and free from available fine-grade material. All reservoirs are, however, not effective sediment sinks. For example, in small, narrow reservoirs with low water residence time sediment deposition may be negligible.

The continuous addition of sediments to reservoirs successively reduces their storage capacity. Small reservoirs in sediment-rich regions can be completely filled within a few decades whereas large reservoirs in rivers running over coarse substrates have an expected life span of several centuries. It has been estimated that almost 30% or more than 100 billion metric tons of the global sediment load is trapped behind dams. This development is one of the factors eliciting demands for construction of new reservoirs. Nutrients, pesticides, and heavy metals that are transported with sediments are also trapped in the reservoirs. Such loadings make it more difficult to restore a river stretch should the dam and the reservoir be regarded as no longer necessary at some time in the future.

Postglacial

The largest external influence on Holocene climate was the 8% decrease in summer radiation (and increase in winter radiation) that occurred between 11 000 years BP and the present due to the precession of the Earth's axis of rotation. Summers were warmer and winters colder over the Northern Hemisphere 11 000 years ago, a contrast that is much reduced today. These changes are expected to have caused stronger monsoon circulation 11 000 years BP, and the strengthening of the monsoon in the early Holocene observed in both data and model simulations is one of the successes of Quaternary paleoclimatology (see Postglacial Indian Ocean).

New proxies have refined our understanding of other climate changes during the postglacial. In the tropics, the ITCZ appears to have been located northward of its present mean position during the early Holocene and appears to have moved south during the Holocene. In the African and Asian sectors, this would have contributed to weakening the monsoon circulation. In the Atlantic, the southward migration of the ITCZ has been documented by decreasing Ti and Fe concentrations in Cariaco Basin sediments, an indicator of decreasing rainfall over the adjacent continent that is best explained by southward movement of the ITCZ during the Holocene (Haug et al., 2001) (see Postglacial North Atlantic). In the Pacific, the east–west SST gradient, indicative of the strength of the Walker circulation, was largest during the mid-Holocene, reduced today, and much reduced during the last glacial interval (Koutavis et al., 2002; see Postglacial North Pacific, Postglacial South Pacific). These changes support the hypothesis of an El-Niño-like mean state in the Pacific during the last glacial interval, and more La-Niña-like mean conditions during the middle Holocene. Progressive freshening of the western Pacific sea surface from the beginning to the end of the Holocene is supported by the oxygen isotope record in the western Pacific, after removal of the temperature effect using Mg/Ca (Stott et al., 2004).

An intriguing aspect of paleoceanographic records of the postglacial interval is the evidence for small-amplitude century-scale cycles that correlate with records of solar variability derived from 10Be and 14C variations. First observed in records of ice-rafting in the North Atlantic (Bond et al., 2001), millennial-scale variability has also been observed in paleoceanographic records of the Asian summer monsoon (Fleitmann et al., 2003; Gupta et al., 2003). Unlike the Milankovitch changes in summer insolation that are large (8%), these changes in (total annual) radiation are small (less than 1%), requiring amplifying mechanisms not yet understood. The changes observed in paleoceanographic records are so small that the changes are close to the uncertainty in most proxies and perhaps only detectable in regions where the changes are unusually large. Some consider the Medieval Warm Period, the Little Ice Age, and part of the warming since the Little Ice Age (during the last century) to be the most recent manifestations of the solar cycles. The societal relevance of this aspect of natural climate variability warrants further research, where paleoceanographic records have great potential.