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Lecture 1: the Earth's Magnetic Field

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Lecture 1: the Earth's Magnetic Field 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 ) r × B = 0 and so there exists a scalar potential V such that B = -rV There are no magnetic monopoles so r · B = 0 and so the scalar potential is harmonic, i.e. r2V = 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 r × B = µoj, where B is the magnetic field, j the current density and µ0 the magnetic permeability. There are no magnetic monopoles so r · B = 0 and so the scalar potential is harmonic, i.e. r2V = 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 r × B = µoj, where B is the magnetic field, j the current density and µ0 the magnetic permeability. In an electrically insulating medium, j = 0 ) r × B = 0 and so there exists a scalar potential V such that B = -rV 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 r × B = µoj, where B is the magnetic field, j the current density and µ0 the magnetic permeability. In an electrically insulating medium, j = 0 ) r × B = 0 and so there exists a scalar potential V such that B = -rV There are no magnetic monopoles so r · B = 0 and so the scalar potential is harmonic, i.e. r2V = 0 Potential field Ampere’s` law r × B = µoj, where B is the magnetic field, j the current density and µ0 the magnetic permeability. In an electrically insulating medium, j = 0 ) r × B = 0 and so there exists a scalar potential V such that B = -rV There are no magnetic monopoles so r · B = 0 and so the scalar potential is harmonic, i.e. r2V = 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. The solution of this equation are of the form l 1 ( θ φ) = l + m(θ φ) V r, , Almr Blm + Yl , 1 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 r2V = 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 r2V = 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 , 1 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.
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