Lecture 1: the Earth's Magnetic Field

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Lecture 1: The Earth’s magnetic ﬁeld

Celine´ Guervilly

School of Mathematics, Statistics and Physics, Newcastle University, UK Outline of these lectures

Lecture 1 Geomagnetic ﬁeld: Sources and spatio-temporal observations

Lecture 2 Dynamo theory, experimental dynamos and numerical simulations

Lecture 3/4 Dynamics of the Earth’s core: rotating ﬂuids and convection Outline of today’s lecture

Sources of the near-Earth magnetic ﬁeld

Magnetic data

Spatial variations of the internal field: core field and crustal field

Temporal variations of the core field on historical and geological timescales Sources of the near-Earth magnetic field Sources of the near-Earth magnetic field

The Earth’s magnetic ﬁeld is the result of the superposition of many ﬁelds produced by a great variety of sources.

Two types of sources: Magnetised media and electric currents.

Magnetised rocks located in the upper layers of the Earth where T < Tc.

Tc: Curie temperature above which materials lose their permanent magnetic properties Iron: Tc = 1043K

Hulot et al. (2015) Field of internal origin

Largest contribution is the core/main ﬁeld 30,000 nT at the equator, 60,000 nT at the poles Changes on secular timescales.

The crustal ﬁeld produced by magnetised rocks (located mainly in the crust). Very dependent on locations: from less than 1nT to 1000nT.

The internal field is dominated by the core field at scales larger than 2000 km and by the crustal field on smaller scales. Hulot et al. (2015) Field of external origin

Electric currents in the ionosphere and the magnetosphere.

Magnitude can strongly vary with time.

“Quiet” times: at the Earth’s surface, magnetospheric ﬁeld about 20 nT and ionospheric ﬁeld about 10 nT (varies with local time, peaks during day time).

“Disturbed” times: external ﬁeld of 1000 nT varying on timescales of less than 1s to several days.

Hulot et al. (2015) Induced ﬁelds

Rocks in the crust and mantle and salty waters in the oceans have a weak electric conductivity: induced currents due to time variations of the main and external ﬁelds.

These induced currents have weak magnitudes, only a small fraction of the inducing ﬁeld.

In the oceans, tides and oceanic flows interacting with the main field produce weak fields of a few nT.

Hulot et al. (2015) Sources of the near-Earth magnetic ﬁeld

A separation of the contributions from each type of ﬁeld can only be achieved only with a large numbers of geomagnetic measurements.

This allows to construct global models of the geomagnetic ﬁeld.

Hulot et al. (2015) Magnetic data Local topocentric coordinate system

X, Y and Z are the components of the ﬁeld pointing towards geographic north, geographic east, and vertically down.

Inclination I: angle between the horizontal plane and the direction of the ﬁeld (positive if downward).

Declination D: angle between the magnetic north and the geographic north (positive if East). Ground data

Network of about 150 ground observatories. Uneven spatial distribution. Good temporal coverage. Accuracy of 1 nT.

Observatories providing data to WDC/INTERMAGNET between 1997 and 2012.

Chambon-La-Foretˆ Ascension Island Antartica Ground data Ground data

Observatory data provided through the INTERMAGNET network and the World Data Center (WDC) system. https://www.ngdc.noaa.gov/geomag/data.shtml Knowledge of the declination is required to use a compass! Satellite data

Global map of the magnetic field. Data are obtained over different regions with the same instrumentation. Measuring from an altitude: reduced the effect of local magnetic heterogeneities. Not directly possible to determine whether observed magnetic field variations are due to temporal or spatial changes.

Magsat, Ørsted, CHAMP (top) and one of the three Swarm satellites (bottom) Spatial variations of the internal ﬁeld Spatial variations of the core ﬁeld at the Earth’s surface

Radial component of the ﬁeld at the Earth’s surface in 2010 (Hulot et al. 2015) Magnetic ﬁeld produced by a dipole

The magnetic ﬁeld produced by a magnetic dipole moment M is

µ M · r B = 0 3 r − M 4πr3 r2

SI unit of M: A m2 −7 µ0 = 4π × 10 H/m is the vacuum magnetic permeability SI unit of the magnetic ﬁeld B: Tesla (T) (cgs unit: Gauss with 1G=10−4T)

The component of an axial dipole in spherical coordinates are

2µ M cos θ B = 0 r 4πr3 µ0M sin θ Bθ = 4πr3 Bφ = 0 Magnetic dipole

Magnetic ﬁeld lines around a bar magnet.

Iron ﬁlings reveal the lines of force around a bar magnet. Spatial variations of the core ﬁeld at the Earth’s surface

Declination at the Earth’s surface in 2010 (Hulot et al. 2015) A pure axial dipole ﬁeld would have zero declination everywhere. Spatial variations of the core ﬁeld at the Earth’s surface

Inclination at the Earth’s surface in 2010 (Hulot et al. 2015) North and South magnetic poles (I = ±90◦) are not antipodal: additional components to the inclined dipole. Spatial variations of the core ﬁeld at the Earth’s surface

Field intensity at the Earth’s surface in 2010 (Hulot et al. 2015) northern maximum (60,000 nT) < southern maximum (66,670 nT). Low-intensity region over South America and the South Atlantic: South Atlantic Anomaly (SAA) In an electrically insulating medium,

j = 0 ⇒ ∇ × B = 0

and so there exists a scalar potential V such that

B = −∇V

There are no magnetic monopoles so

∇ · B = 0

and so the scalar potential is harmonic, i.e.

∇2V = 0

Assuming that the conductivity of the mantle and crust is negligible and the permanent magnetisation of the crust is very weak, the scalar potential description is valid from the CMB up to the base of the ionosphere. A solution to Laplace’s equation is uniquely determined if the value of the function or its normal derivative is speciﬁed on all boundaries.

Potential ﬁeld

Ampere’s` law ∇ × B = µoj,

where B is the magnetic ﬁeld, j the current density and µ0 the magnetic permeability. There are no magnetic monopoles so

∇ · B = 0

and so the scalar potential is harmonic, i.e.

∇2V = 0

Potential ﬁeld

Ampere’s` law ∇ × B = µoj,

where B is the magnetic ﬁeld, j the current density and µ0 the magnetic permeability. In an electrically insulating medium,

j = 0 ⇒ ∇ × B = 0

and so there exists a scalar potential V such that

B = −∇V Assuming that the conductivity of the mantle and crust is negligible and the permanent magnetisation of the crust is very weak, the scalar potential description is valid from the CMB up to the base of the ionosphere. A solution to Laplace’s equation is uniquely determined if the value of the function or its normal derivative is speciﬁed on all boundaries.

Potential ﬁeld

Ampere’s` law ∇ × B = µoj,

where B is the magnetic ﬁeld, j the current density and µ0 the magnetic permeability. In an electrically insulating medium,

j = 0 ⇒ ∇ × B = 0

and so there exists a scalar potential V such that

B = −∇V

There are no magnetic monopoles so

∇ · B = 0

and so the scalar potential is harmonic, i.e.

∇2V = 0 Potential ﬁeld

Ampere’s` law ∇ × B = µoj,

where B is the magnetic ﬁeld, j the current density and µ0 the magnetic permeability. In an electrically insulating medium,

j = 0 ⇒ ∇ × B = 0

and so there exists a scalar potential V such that

B = −∇V

There are no magnetic monopoles so

∇ · B = 0

and so the scalar potential is harmonic, i.e.

∇2V = 0

l 1 ( θ φ) = l + m(θ φ) V r, , Almr Blm + Yl , ∞ rl 1 = Xl=1 mX0 m where the spherical harmonics Yl of degree l and order m is deﬁned as

m m m imφ Yl (θ, φ) = Cl Pl (cos θ)e

m m where Cl is a normalisation constant and Pl is the associated Legendre polynomials. Spherical harmonics constitute a basis of diﬀerentiable functions. Orthogonality: m n Yl Yk dS = 0 if k , l or n , m ZS Normalisation:

/ 1 (2l + 1)(l − m)! 1 2 YmYmdS = 1 for Cm = (−1)m 4π l l l (l + m)!) ZS

Spherical harmonics decomposition

In spherical coordinates,

1 ∂ ∂V 1 ∂ ∂V 1 ∂2V ∇2V = r2 + sin θ + = 0 r2 ∂r ∂r r2 sin θ ∂θ ∂θ r2 sin2 θ ∂2φ Spherical harmonics constitute a basis of diﬀerentiable functions. Orthogonality: m n Yl Yk dS = 0 if k , l or n , m ZS Normalisation:

/ 1 (2l + 1)(l − m)! 1 2 YmYmdS = 1 for Cm = (−1)m 4π l l l (l + m)!) ZS

Spherical harmonics decomposition

In spherical coordinates,

1 ∂ ∂V 1 ∂ ∂V 1 ∂2V ∇2V = r2 + sin θ + = 0 r2 ∂r ∂r r2 sin θ ∂θ ∂θ r2 sin2 θ ∂2φ The solution of this equation are of the form

l 1 ( θ φ) = l + m(θ φ) V r, , Almr Blm + Yl , ∞ rl 1 = Xl=1 mX0 m where the spherical harmonics Yl of degree l and order m is deﬁned as

m m m imφ Yl (θ, φ) = Cl Pl (cos θ)e

m m where Cl is a normalisation constant and Pl is the associated Legendre polynomials. Spherical harmonics decomposition

In spherical coordinates,

1 ∂ ∂V 1 ∂ ∂V 1 ∂2V ∇2V = r2 + sin θ + = 0 r2 ∂r ∂r r2 sin θ ∂θ ∂θ r2 sin2 θ ∂2φ The solution of this equation are of the form

l 1 ( θ φ) = l + m(θ φ) V r, , Almr Blm + Yl , ∞ rl 1 = Xl=1 mX0 m where the spherical harmonics Yl of degree l and order m is deﬁned as

m m m imφ Yl (θ, φ) = Cl Pl (cos θ)e

/ 1 (2l + 1)(l − m)! 1 2 YmYmdS = 1 for Cm = (−1)m 4π l l l (l + m)!) ZS Spherical harmonics decomposition

l = 6, m = 0 l = 16, m = 9 l = 9, m = 9 Gauss coeﬃcients

Internal solutions (∝ 1/rl+1):

l a l+1 V = a gm cos(mφ) + hm sin(mφ) Pm(cos θ) r l l l = Xl mX0 with a the Earth’s radius. International Geomagnetic Reference Field (IGRF)

Every 5 years, the International Association of Geomagnetism and Aeronomy (IAGA) publishes a spherical harmonic model of the magnetic field at the Earth’s surface calculated from magnetic data. IGRF-12: Predictive model for 2015-2020, Definitive Geomagnetic Reference Field for 1945-2010. Gauss coefficients provided up to degree l = 13. IGRF available on IAGA website https://www.ngdc.noaa.gov/IAGA/vmod/igrf.html. International Geomagnetic Reference Field (IGRF) International Geomagnetic Reference Field (IGRF) International Geomagnetic Reference Field (IGRF) ∂V 2a3 B = − = g0 cos θ r ∂r r3 1 3 1 ∂V a 0 Bθ = − = g sin θ r ∂θ r3 1 1 ∂V Bφ = − = 0 r sin θ ∂φ Comparing with the components of an axial dipole, we get: µ M g0 = 0 1 4πa3

0 g1: axial dipole 1 ◦ g1: equatorial dipole at 0 longitude 1 ◦ h1: equatorial dipole at 90 longitude 0 g2: axial quadrupole

Physical meaning of the Gauss coeﬃcients

l a l+1 V = a gm cos(mφ) + hm sin(mφ) Pm(cos θ) r l l l = Xl mX0 For l = 1, m = 0: a3 a3 V = g0P0(cos θ) = g0 cos θ r2 1 1 r2 1 Comparing with the components of an axial dipole, we get: µ M g0 = 0 1 4πa3

0 g1: axial dipole 1 ◦ g1: equatorial dipole at 0 longitude 1 ◦ h1: equatorial dipole at 90 longitude 0 g2: axial quadrupole

Physical meaning of the Gauss coeﬃcients

l a l+1 V = a gm cos(mφ) + hm sin(mφ) Pm(cos θ) r l l l = Xl mX0 For l = 1, m = 0: a3 a3 V = g0P0(cos θ) = g0 cos θ r2 1 1 r2 1

∂V 2a3 B = − = g0 cos θ r ∂r r3 1 3 1 ∂V a 0 Bθ = − = g sin θ r ∂θ r3 1 1 ∂V Bφ = − = 0 r sin θ ∂φ Physical meaning of the Gauss coeﬃcients

l a l+1 V = a gm cos(mφ) + hm sin(mφ) Pm(cos θ) r l l l = Xl mX0 For l = 1, m = 0: a3 a3 V = g0P0(cos θ) = g0 cos θ r2 1 1 r2 1

∂V 2a3 B = − = g0 cos θ r ∂r r3 1 3 1 ∂V a 0 Bθ = − = g sin θ r ∂θ r3 1 1 ∂V Bφ = − = 0 r sin θ ∂φ Comparing with the components of an axial dipole, we get: µ M g0 = 0 1 4πa3

0 g1: axial dipole 1 ◦ g1: equatorial dipole at 0 longitude 1 ◦ h1: equatorial dipole at 90 longitude 0 g2: axial quadrupole CMB

Downward continuation of the core ﬁeld to the CMB for l < 13 as a potential ﬁeld.

Lowes-Mauersberger spatial power spectrum

The power spectra of the magnetic ﬁeld at a radius r is

n 2l+4 a h m 2 m 2i D 2E Rl(r, t) = (l + 1) gl (t) + hl (t) and kB(r, θ, φ, t)k = Rl(r, t) r ∞ = mX0 Xl=1

Earth’s surface Spectrum dominated by the core ﬁeld at degrees l < 14 and by the crustal ﬁeld for l > 14 at the Earth’s surface. Lowes-Mauersberger spatial power spectrum

The power spectra of the magnetic ﬁeld at a radius r is

n 2l+4 a h m 2 m 2i D 2E Rl(r, t) = (l + 1) gl (t) + hl (t) and kB(r, θ, φ, t)k = Rl(r, t) r ∞ = mX0 Xl=1

Earth’s surface CMB Spectrum dominated by the core field at degrees l < 14 and by the crustal field for l > 14 at the Earth’s surface. Downward continuation of the core field to the CMB for l < 13 as a potential field. Spatial variation of the magnetic field at the core surface

Radial component of the field at the CMB in 2010 (Hulot et al. 2015) Field is more complicated than at the surface, because higher spherical harmonics are amplified more (by a factor (a/c)l+2 where c is the core radius). Reversed flux patches near the geographic north pole and in the southern hemisphere (related to the SAA observed at the surface). High-intensity lobes near the polar regions. Crustal field

Total field anomaly map from EMAG2 model (Maus et al., 2009). Strong spatial variability of the crustal field due to the history of its magnetised sources and from the various magnetisation properties of rocks. Spatial spectrum of the crustal field is characterised by comparable contributions at all length scales. Temporal variations of the core field 1. Historical record and secular variation Secular variation at the CMB

Secular variation of the radial component of the ﬁeld, ∂tBr, at the CMB in 2010 up to l = 13 (Hulot et al. 2015) Relatively weak secular variation in the Paciﬁc hemisphere. Historical records of the secular variation

(Jackson & Finlay 2015) Historical records of the secular variation

Number of historical data used in the construction of gufm1 (Jackson et al., 2000) A large part of the dataset originates in marine observations of the declination (taken for navigation purposes). Position is part of the uncertainty of magnetic measurements: no problem on lands, but more challenging at seas. Historical records of the secular variation

Geographic data distribution of declination observations made before 1590 (n = 160) (Jackson & Finlay 2015) Historical records of the secular variation

Geographic data distribution of declination observations made between 1590 and 1699 (n = 12001) (Jackson & Finlay 2015) Historical records of the secular variation

Geographic data distribution of declination observations made between 1700 and 1799 (n = 68076) (Jackson & Finlay 2015) Historical records of the secular variation

Geographic data distribution of declination observations made between 1800 and 1930 (n = 71323) (Jackson & Finlay 2015) Wandering of the magnetic poles

Observed magnetic pole in the north hemisphere during 1831-2007 (yellow squares) and modelled pole locations from 1590 to 2020 (circles from blue to yellow). Wandering of the magnetic poles

Northward velocity of the magnetic poles in the northern (purple) and southern (orange) hemispheres (Thebault´ et al. 2015) Westward drift and hemispherical asymmetry

In 1590, the line D = 0 bisected Africa. In 1990, this line bisects South America.

Declination at the Earth’s surface in (a) 1590 and (b) 1990 (from gufm1 Jackson et al 2000) Westward drift and hemispherical asymmetry

Maxima and minima of inclination anomalies centered on low latitudes can be tracked westward.

Inclination anomaly at the Earth’s surface in (a) 1590 and (b) 1990 (from gufm1 Jackson et al 2000) Westward drift and hemispherical asymmetry

Vertical component of the ﬁeld (Z) at the Earth’s surface (from gufm1 Jackson et al 2000). Drift of the South Atlantic Anomaly

Location (a) and values (b) of the minimum intensity at the Earth’s surface (Pavon-Carrasco´ & De Santis 2016) Decay of the axial dipole

(Jackson & Finlay 2015) Axial dipole has decayed at an average rate of 5% per century since 1840. Geomagnetic jerks (secular variation impulses)

Abrupt changes in the second time derivative of the main field. Separate intervals of linearly changing secular variation Identified as having occurred in 1901, 1913, 1925, 1969, 1978, 1991, and 1999. Jerks are not always observed at all locations and not always simultaneous. Physical process is not well understood. Secular variation of the Eastward component of the field at Niemegk Unpredictable occurence: difficult for observatory (Germany) IGRF predictive secular variation (Jackson & Finlay 2015) models to predict the main field evolution (even over 5 years). Low-latitude westward drifting field features at the CMB

movie

Time-longitude plots of the non-axisymmetric Br at the CMB (Finlay & Jackson, 2003).

Westward motion (speed ∼ 17 km/year) at low latitudes: Hydromagnetic wave propagation? Advection by an azimuthal flow near the core surface? Temporal variations of the core field 2. Archeo and Paleomagnetic record: Variations on millennial and geological timescales Magnetic data must be combined with an age estimate (absolute chronological methods, stratigraphic, specific style of pottery, eye-witness account of a volcanic eruption...). Problem for paleomagnetic direction data if a structure, sediment, or lava flow has been disturbed after magnetisation: need to know the relationship between the coordinate system in which measurements are made and the geographic system in which the magnetisation was originally acquired.

Magnetic data on millennial timescales

Ambient magnetic field recorded in lavas and archeological artifacts (kilns, ceramics...) after cooled down below the Curie temperature. Magnetised particles oriented in the direction of the ambient field in lake sediments. Problem for paleomagnetic direction data if a structure, sediment, or lava flow has been disturbed after magnetisation: need to know the relationship between the coordinate system in which measurements are made and the geographic system in which the magnetisation was originally acquired.

Magnetic data on millennial timescales

Ambient magnetic field recorded in lavas and archeological artifacts (kilns, ceramics...) after cooled down below the Curie temperature. Magnetised particles oriented in the direction of the ambient field in lake sediments. Magnetic data must be combined with an age estimate (absolute chronological methods, stratigraphic, specific style of pottery, eye-witness account of a volcanic eruption...). Problem for paleomagnetic direction data if a structure, sediment, or lava flow has been disturbed after magnetisation: need to know the relationship between the coordinate system in which measurements are made and the geographic system in which the magnetisation was originally acquired. Magnetic data on millennial timescales

Numbers of (a) archeomagnetic and lava and (b) sediment paleomagnetic data (Constable & Korte 2015) Magnetic data on millennial timescales

Location of archeomagnetic and lava data sites (left) and sediment record (right), for declination (top), inclination (middle) and intensity (bottom) (Constable & Korte 2015) Weak constraint over oceans and in the southern hemisphere. Quasi-stationary high-latitude magnetic ﬂux patches in both hemispheres. Millennial models have lower intensity in the south than in the north: consequence of data distribution or genuine persistent north-south hemispheric asymmetry? The South Atlantic Anomaly is absent in the millennial models: lowest intensity regions near Indonesia and Australia.

Geomagnetic ﬁeld variations on millennial timescales

Intensity at the Earth’s surface (left) and Br at the CMB (right) for models averaged over 400, 3000, and 10000 years (top to bottom) (Constable & Korte 2015) Millennial models have lower intensity in the south than in the north: consequence of data distribution or genuine persistent north-south hemispheric asymmetry? The South Atlantic Anomaly is absent in the millennial models: lowest intensity regions near Indonesia and Australia.

Geomagnetic ﬁeld variations on millennial timescales

Intensity at the Earth’s surface (left) and Br at the CMB (right) for models averaged over 400, 3000, and 10000 years (top to bottom) (Constable & Korte 2015) Quasi-stationary high-latitude magnetic ﬂux patches in both hemispheres. The South Atlantic Anomaly is absent in the millennial models: lowest intensity regions near Indonesia and Australia.

Geomagnetic ﬁeld variations on millennial timescales

Intensity at the Earth’s surface (left) and Br at the CMB (right) for models averaged over 400, 3000, and 10000 years (top to bottom) (Constable & Korte 2015) Quasi-stationary high-latitude magnetic ﬂux patches in both hemispheres. Millennial models have lower intensity in the south than in the north: consequence of data distribution or genuine persistent north-south hemispheric asymmetry? Geomagnetic ﬁeld variations on millennial timescales

Intensity at the Earth’s surface (left) and Br at the CMB (right) for models averaged over 400, 3000, and 10000 years (top to bottom) (Constable & Korte 2015) Quasi-stationary high-latitude magnetic ﬂux patches in both hemispheres. Millennial models have lower intensity in the south than in the north: consequence of data distribution or genuine persistent north-south hemispheric asymmetry? The South Atlantic Anomaly is absent in the millennial models: lowest intensity regions near Indonesia and Australia. Dipole moment on millennial timescales

Evolution of the geomagnetic dipole moment over the last 10,000 years (Constable & Korte 2015) In the last 10000 years, the dipole moment was highest between 1000 BC-AD 1000. The dipole moment was about 30% lower than the maximum values between 7000-3000 BC. The current decrease in dipole moment does not appear anomalous compared with other variations: a geomagnetic reversal might not be imminent. Dipole tilt on millennial timescales

Evolution of the tilt of the dipole axis over the last 10,000 years (Constable & Korte 2015)

Large uncertainties and agreement between models is variable over time. The present dipole tilt is not particularly large if uncertainty estimates are taken into account. 1350-year cycle in the variation of the dipole axis? (Nilsson et al. 2011) Seaﬂoor spreading

Age of the oceanic crust (red: youngest, blue: oldest (180Myr)) The rocks of the oceanic crust contain magnetite. The rocks acquire a remanent magnetisation aligned with the ambiant geomagnetic ﬁeld at the time of cooling. Spreading velocity = 2.5 cm/year = 25km/million year Geomagnetic polarity over the last 5 million years Black = normal polarity, White = reversed polarity

Magnetic anomalies on the sea ﬂoor Magnetic anomalies on the sea ﬂoor

Geomagnetic polarity over the last 5 million years Black = normal polarity, White = reversed polarity Geomagnetic polarity over the last 160 Myr

Last polarity reversal: 780,000 years ago Reversals occur at random intervals: 0.1-50 Myr. On average, a few times every Myr. Polarity reversals

Valet et al (2005) The Virtual Axial Dipole Moment (VADM) is the equivalent geocentric axial dipole moment that would produce the observed intensity at the latitude of the data sample. The axial dipole begins to decay 60,000 years before the reversal and rebuilds itself in only a few 1000 years. Paleointensities and excursions

Guyodo & Valet (1999) Geomagnetic excursions are observed when the dipole moment decreases below 4 × 1022 A m2. Aborted reversals? Precambrian paleointensities

Valet (2003) Geomagnetic ﬁeld present for at least 3.5Gyr (Tarduno et al 2010) Oldest measurements: magnetic inclusions in silicate crystals.