Plasma sheet
Magnetotail Magnetosheath
Inner radiation belt
Outer radiation belt
Magnetopause Bow shock
Dayside Nightside
Solar wind
10 0
-10 20
Distance [Earth radii]
Figure 2.3: Illustration of the Earth’s magnetosphere.
All the re-calibrated sunspot number series described above are reconstructed from existing data sets and it is not trivial to determine which series describes the reality best.
With the implementation of the sunspot numbers in reconstructing the solar irradiance (Sect. 1.4.2), the choice of input series with an increasing long-term trend (Hoyt and Schatten 1998;Usoskin et al. 2016c) or a rather constant long-term trend (Cliver and Ling 2016;Svalgaard and Schatten 2016) leads to different TSI/SSI reconstructions (Kopp et al.
2016;Lockwood et al. 2016).
However, since the major differences among all the proposed revised SN series lie between the 18th – 19th centuries while the overall solar activity of the 17th and the 20th centuries are in agreement, the choice of the sunspot number series does not affect the secular trend in solar activity from the Maunder minimum to the modern maximum (Kopp 2016;Wu et al. 2018a).
2.2 Geomagnetic field
The Chinese discovered, more than 1 000 years ago, that a needle always aligns to the di-rection of north-south when floating on water. Since lodestone is a naturally magnetized piece of the mineral magnetite, this discovery eventually led to the invention of the mo-dern compass. Soon after, the compass was widely used for navigation and has increased the speed of civilisation all around the world. In 1600, William Gilbert first described the Earth as a giant magnet. Two centuries later, a mathematical model was developed by Carl Friedrich Gauß, describing the source of the majority of the measured magnetic field origins from the interior of the Earth. The Earth’s magnetic field is briefly introduced in the following.
The Earth’s magnetic field, also called the geomagnetic field, was later found to be generated in the Earth’s interior through a self-sustaining dynamo process in the Earth’s core, which consists of electrical currents driven by rotation and convection of the viscous material (e.g.,Glatzmaier and Roberts 1995; Glatzmaier et al. 2004; Olson et al. 1999;
Buffett 2000). The geomagnetic field extends outwards from the interior of the Earth into interplanetary space. This field surrounds and protects the Earth from energetic charged particles. The general shape of the geomagnetic field can be approximated by a dipole.
However, due to the solar wind, the geomagnetic field is compressed at the dayside and extended at the nightside. Therefore, the shape of the geomagnetic field appears like a comet-tail, as illustrated in Fig. 2.3. The dayside magnetopause is at about 10 Earth’s radii (RE) and the tail of the magnetosphere extends to at least 100RE(Axford et al. 1965;
Dessler and Juday 1965;Ness 1969).
The geomagnetic field can be described with three orthogonal components at any given point on the Earth’s surface. One of the common systems is a Cartesian coordinate system, (X,Y,Z), where X points geographically northwards, Y points eastwards and Z points downwards (into the Earth). The magnetic field vector at any point is described as Bwith magnetic field intensity B. H is the projection ofB on the horizontal plane with an intensityH, defined as
√
X2+Y2, as shown in Fig. 2.4. An alternative common way to specify any given point on Earth is by its total magnetic intensityB and two angular quantities: declination (D) and inclination (I). The declination is the angle between the horizontal direction and the geographic north,D=tan−1 YX
. The inclination is defined as the angle between the total field vectorBand the horizontal plane,I = tan−1 HZ
.
Since the magnetic fieldB is a conservative force (i.e., ∇ ×B = 0), the field can be expressed as a gradient of a scalar potentialV as
B= −∇V. (2.3)
Using the solenoidal characteristic of the magnetic field (∇ ·B = 0, i.e., magnetic monopoles do not exist), we can derive Eq. (2.3) as a Laplace’s equation:
∇2V =0. (2.4)
The scalar potential of the geomagnetic fieldV is a function of radius r, co-latitude θ, and longitudeφ(Chapman and Bartels 1940). By assuming the Earth is a sphere (and ignoring the external source term of the magnetic field which is small and transient), the solution of Laplace’s equation (Eq. 2.4) is usually described by spherical harmonic analysis (SHA): wherePml are the Schmidt quasi-normalized Legendre functions of degreeland orderm (e.g.,Backus et al. 1996). Thegml andhml are known as Gaussian coefficients, which are determined by fitting observational data to Eq. (2.5). The geomagnetic field is composed of a main magnetic dipole component and other small non-dipole components, described by the Gaussian coefficients. The magnitude of the Gaussian coefficients decreases with increasing order, i.e., the higher order the coefficients, the less significant contribution to the whole magnetic field.
2.2 Geomagnetic field Geographic North
Y East X
Z
B D
I
H
Down
Figure 2.4: Sketch of coordinates used to describe magnetic fields.Bis the vector repre-senting the magnetic field of the earth with magnitudeB. His the projection of the field, B, onto the horizontal plane. X, Y and Z represent the northward, eastward, and down-ward directions, respectively. Declination,D, is the angel betweenHand the geographic north. I, inclination, is the angle betweenBand the horizontal plane.
The geomagnetic field is not a static field. Instead, it varies significantly with time throughout the Earth’s history. There have been discoveries of the changes in the strength and direction of the geomagnetic field. One of the extreme changes in the geomagnetic field history is the geomagnetic reversal. Evidence for these events can be found near the mid-ocean ridges, where the rocks were magnetized when the lava temperature cools down and pushed aside by new volcanic activities, and the magnetization is preserved until today. The reversal has a pattern of occuring on average every few hundred thousand years (Gubbins 1994; Merrill and McFadden 1999; Rüdiger and Hollerbach 2004) with reversal durations lasting from a few thousand up to 28 000 years (Clement 2004). Yet the mechanism of reversal is still a mystery. From the remnant magnetization measurements in ocean sediments and lava flows, geologists have established a last full reversal event, the Brunhes–Matuyama event. The Brunhes–Matuyama event occurred at about 778 000 ± 2000 years ago with a duration of about 12 000 years (Singer and Pringle 1996;Gradstein et al. 2005). From the last full reversal till now, there was only one short event confirmed.
TheLaschamp excursion, which took place about 41 000 ±2 000 years ago, was a short and not complete reversal. The Laschamp excursion can be seen in many geological archives as well as from the long10Be flux records in ice cores, such as GRIP (Greenland Ice Core Project) and GISP (Greenland Ice Sheet Project), which show an increase in flux by a factor of 2 during this period (see Fig. 1 inMuscheler et al. 2005).
On shorter time scales, the location of the geomagnetic pole has also been discovered to be varying. This phenomenon is known aspolar wander. A recent study shows that
the geomagnetic north pole moves quickly (tens of kilometres per year) towards the north (McElhinny and McFadden 2000). The magnetic north pole, which was in Siberia at about 1000AD, had moved to its presents position in northern Canada (see Fig. 5.8.1-1 in Beer et al. 2012). The geomagnetic poles are offset from the rotation axis by ≈11◦ and the closest distance between the centre of mass of the Earth and the magnetic axis is
≈450 km.
To estimate the characteristics of the geomagnetic field in the past, one requires the information collected from either archaeology or geology. These two corresponding met-hods are termed “archaeomagnetic studies” and “paleomagnetic studies”, respectively (McElhinny and McFadden 2000). Both methods are built based on a common fact: under certain conditions, the magnetic elements in the rocks align with the local geomagnetic field and the degree of magnetization is proportional to the geomagnetic field strength at that point of time. After some time, the magnetization is locked in the rocks and remains even if the local geomagnetic field changes. By measuring the strength and the direction of the magnetization in rocks, the properties of the corresponding local magnetic field at any given point of time can be determined. Based on this concept, as long as one deter-mines the time when the magnetization of the rocks was created and locked, the varying geomagnetic field in the past can be reconstructed.
The first method of determining the time when the rock was magnetized is through ar-chaeological sampling. When early humans used fire or made goods out of clay by firing it, the magnetization of the rocks was reset during this process. Due to human activity throughout history, archaeomagnetic studies provide a large amount of data and informa-tion about the past 10 000 years. The second method of determining the magnetizainforma-tion time is by analysing rocks formed via natural geological processes. The most common example is the magnetization through volcanic activities. The rocks lose their original permanent magnetic properties when its temperature reaches the Curie temperature (TC) of the corresponding material and then is reset according to the local geomagnetic field when the rocks cool down. In addition, the magnetic moments in the sediments align with the local geomagnetic field and are locked when the sediments reach a certain de-gree of compaction. This method allows for investigation back to≈1 000 000 years ago.
It is important to note that both methods have uncertainties either caused by human acti-vity (e.g., relocation of the goods and rocks) or by natural events (e.g., earthquake and subsidence). Furthermore, certain chemical reactions caused by lightning can change the characteristics of the magnetization in the rocks.
However, in most paleomagnetic studies, the data sampling is not sufficient to recon-struct the complex magnetic recon-structure. Therefore, the evolution of Earth’s magnetic field is often simply described by an axial dipole. Virtual dipole moment (VDM,Korte and Constable 2005, 2006) and Virtual axial dipole moment (VADM,McElhinny and Sena-nayake 1982;Yang et al. 2000;Usoskin et al. 2016a) are often used to approximate the true dipole moment (DM) and to describe the magnetic field from a limited amount of available archaeo- and palaeomagnetic data (Constable 2007). The VDM is based on the assumption that the field originates only from a geocentric dipole (i.e., the centre of the geomagnetic dipole is located the centre of the Earth and the axis aligns the true magnetic axis) while the VADM assumes the field originates from a geocentric axial dipole (i.e., the centre of the geomagnetic dipole is located at the centre of the Earth and the axis aligns with the rotational axis). However, such simple assumptions about the magnetic structure
2.3 Cosmic rays