Chapter 4 Multipole model of the Earth's magnetic field

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1 Chapter 4 Multipole model of the Earth's magnetic field 1

2 Previously A measurement of the geomagnetic field at any given point and time consists of a superposition of fields from different sources: Internal sources: Crustal or anomalous or lithospheric field: The magnetic field caused by magnetized rocks in the lithosphere (< 50 km depth) can locally exceed the strength of the Earth's main field, but globally constitutes <1% of the field. External sources: Core or main field: A hydrodynamic dynamo in the Earth's fluid outer core ( km depth) produces over 99% of the Earth's magnetic field. Solar activity drives electric currents in the Earth's ionosphere and magnetosphere (> 100 km altitude) which cause irregular magnetic field variations with periods from seconds to hours. The first order approximation of the Earth's rather complex magnetic field is the dipole model.

3 Content Spherical harmonic expansion of the Earth's magnetic field Properties of the spherical harmonic expansion International Geomagnetic Reference Field (IGRF) 3

4 Multipole model of the Earth's magnetic field Generalization of the dipole model of the Earth's magnetic field. Basic assumption: the Earth's magnetic field can be represented as a superposition of the fields created by several multipole magnets located at the center of the Earth. The simplest multipole magnet is the dipole, then quadrupole (four poles), octupole (eight poles), etc. 4

5 Spherical harmonics Spherical harmonics are a series of special functions defined on the surface of a sphere. As Fourier series are a series of functions used to represent functions on a circle, spherical harmonics are a series of functions that are used to represent functions defined on the surface of a sphere. Spherical harmonics are defined as the angular portion of a set of solutions to Laplace's equation in three dimensions. Spherical harmonics are functions defined in terms of spherical coordinates and and organized by wavelength. Y lm (θ, ϕ)= l +1 (l m)! m P l (cos θ) ei m ϕ 4 π (l + m)! θ co-latitude ϕ longitude l degree of spherical harmonic m order of spherical harmonic P ml (cos θ) associated Legendre function of the first kind 5

6 Legendre functions The first few Legendre functions P n ( x) are: P 0 ( x)=1 P 1 ( x)=x 1 P ( x)= (3x 1) 1 3 P 3 ( x)= (5x 3x) 6

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8 Unnormalized associated Legendre functions of the first kind The unnormalized associated Legendre functions P mn ( x ) are related to the Legendre functions P n ( x) by: 0 m=0: P n ( x)= P n ( x) m m m m / d m 0: P n ( x)=( 1) (1 x ) m P n ( x) dx Matlab function: P = legendre(n,x) 8

9 The first few unnormalized associated Legendre functions P mn ( x) are: 0 P 0 ( x)=1 0 P 1 ( x)= x 1 P 1 ( x)= (1 x )1/ 1 0 P ( x)= (3x 1) 1 P ( x)= 3x(1 x )1/ P ( x)=3(1 x ) P 3 ( x)= (5x 3x) 3 1 P 3 ( x)= (5x 1)(1 x )1 / P 3 ( x)=15x (1 x ) P 33 ( x)= 15(1 x )3 / 9

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11 Visual representations of the first few real spherical harmonics. Blue portions represent regions where the function is positive, and yellow portions represent where it is negative. The distance of the surface from 11 m the origin indicates the value of Yl (θ,φ) in angular direction (θ,φ).

12 Wavelength related to harmonic degree Spherical harmonics have n-m zeros on parallels in π radians of co-latitude m zeros on meridians in π radians of longitude. Spherical harmonics with m = 0 are called zonal, i.e., the functions are independent of the longitude φ. Spherical harmonics with m = l are called sectorial, i.e., they represent bands of longitude. Spherical harmonics with m l 0 are called tesseral. 1

13 Although the spherical harmonics are functions on a two-dimensional surface, it is sometimes convenient to characterize them by a one-dimensional wavelength λ. Since a spherical harmonic has n zeros on π radians, λ is taken to be: λ= π R E sin θ n n=6 π λ 13

14 Laplace equation B=0 B= U, where U is the scalar potential. B=0 U =0 (Laplace equation) 1 U 1 U 1 U= r + sin θ + =0 θ θ r r r r sin θ r sin θ ϕ ( ) ( ) There are two types of solutions: - potential U i due to sources internal to the Earth (r < R E ) - potential U e due to sources external to the Earth (r > R E ) such that U =U i +U e. 14

15 The solutions are given as multipole or spherical harmonic expansions: n+ 1 n RE U i (r,θ, ϕ, t)=r E ( g mn (t )cos m ϕ +h mn (t )sin m ϕ) SP mn (cos θ) r n=1 m=0 [( ) [( ) U e (r, θ, ϕ, t)= R E n=1 RE r n n (q mn (t) cos m ϕ+ s mn (t )sin m ϕ )SP mn (cos θ) m=0 ] ] r radius R E Earth radius ( R E =6371. km) θ co-latitude ϕ longitude t time n n degree of multipole (a multipole of degree n has poles) m order of multipole g, h, q, s spherical harmonic coefficients (describe the strength of the multipole magnet in nt) SP mn (cos θ) Schmidt semi-normalized associated Legendre function of the first kind 15

16 Schmidt semi-normalized associated Legendre functions m Schmidt semi-normalized associated Legendre functions SP n ( x ) are related to the unnormalized associated Legendre functions P mn ( x ) by: m=0: SP 0n ( x)= P 0n ( x )=P n ( x ) m m m 0: SP n ( x)=( 1) (n m)! m Pn ( x ) (n+ m)! Matlab function: P = legendre(n,x,'sch') 16

17 The first few Schmidt semi-normalized associated Legendre functions SP mn ( x) are: SP 00 ( x)=1 SP 01 ( x)= x SP 11 ( x )=(1 x )1/ 1 SP 0 ( x )= (3x 1) SP 1 ( x )= 3 x (1 x )1/ 3 SP ( x )= (1 x ) 1 SP 03 ( x)= (5x 3 3x) 6 SP 13 ( x )= (5x 1)(1 x )1/ 4 15 SP 3 ( x )= x (1 x ) 10 (1 x )3/ SP 33 ( x )= 4 17

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19 Spherical harmonic expansion of the magnetic field ( B= U = U 1 U 1 U e r + e θ + e r r θ r sin θ ϕ ϕ Sources internal to the Earth: Ui n RE B r (r,θ, ϕ, t )= = (n+1) r n=1 r max i [ ( ) [( ) [( ) n max 1 Ui Bθ (r,θ, ϕ, t)= = r θ n=1 i RE r n max 1 Ui Bϕ (r,θ, ϕ,t )= = r sin θ ϕ n=1 i Sources external to the Earth: n Ue RE B r (r, θ, ϕ, t )= = n r r n=1 n 1 U e Bθ (r,θ, ϕ, t )= = r θ n=1 max e RE r n max 1 U e Bϕ (r, θ, ϕ, t )= = r sin θ ϕ n=1 e n+ n ( g mn (t) cos m ϕ+hmn (t)sin m ϕ) SP mn (cos θ) m=0 n+ dsp mn (cos θ) ( g (t )cos m ϕ+ h (t)sin m ϕ) dθ m =0 n m n n+ RE r n+1 m n m n (qmn (t)cos m ϕ + smn (t)sin m ϕ) SP mn (cos θ) m=0 dsp mn (cos θ) (q (t)cos m ϕ + s (t)sin m ϕ) dθ m=0 n+1 m n m n ] ] n m n RE r ] m n n n+1 ] msp mn (cos θ) ( g (t )sin m ϕ+h (t )cos m ϕ) sin θ m=0 n [( ) [( ) [( ) max e ) ] ] msp n (cos θ) ( q (t)sin m ϕ + s (t )cos m ϕ) 19 sin θ m=0 m n m n

20 In practice, the sum to infinity has to be truncated at n = nmax, determined by the number of available observations. There are nmax + nmax coefficients in the expansion. In practice, the number of observations is generally much larger than this. The values of the coefficients are determined using the least squares method. 0

21 d m SP n (cos θ) from recurrence formulas dθ m>0 : d m m 1 m P n ( x)=(n+ m)(n m+1) 1 x P n ( x )+m x P n ( x ) dx (n m)! d m d SP mn ( x)=( 1)m P ( x) dx (n+m)! dx n d P mn (cos θ) d cos θ d P mn (cos θ) d P mn (cos θ) = = sin θ dθ dθ d (cos θ) d (cos θ) (1 x ) m=0 : d P ( x)=np n 1 ( x) nxp n ( x) dx n d d SP n ( x)= P n ( x ) dx dx d P n (cos θ) d cos θ d P n (cos θ) d P n (cos θ) = = sin θ dθ dθ d (cos θ) d (cos θ) (1 x ) 1

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23 Properties of the spherical harmonic expansion 3

24 In case of the internal field: Large wavelengths (n < 14, approximately) are associated with the main field: n = 1: dipole component n < 14: anomalous or non-dipole components (Note: The terms n < 14 are considered anomalous relative to the dipole field whereas the terms n > 14 representing the crustal field are considered anomalous relative to the main field. Thus, the anomalous component is always relative, and the reference level should be mentioned.) Smaller wavelengths (n > 14) are associated with the magnetic anomalies of the crust. 4

25 Average magnetic field In case of the magnetic field due to internal sources, the average of B mn squared over the Earth's surface S is defined as: 1 ( B mn ) = ( B mn ) ds 4π It can be shown that for each multipole ( n ) the result is: n ( B n ) =(n+1) ( g mn ) +(h mn ) m=0 This formula can be used to estimate the relative strengths of the different multipoles: (Model: POMME ) n ( B ) 100 ( B ) n 1 ( B ) 100 % ( B ) n N n =1 n

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27 7 Minimum wavelength at the equator (θ = 90o) associated with the spherical harmonic of degree n.

28 If r =RC < R E (source region of the magnetic field at the surface of the liquid core): ( n+ ) RE ( B n) C = ( B n ) E RC R log 10 ( ( B n ) E )= (n+) log10 ( E )+log 10 ( ( B n ) C ) RC RE k = log10 ( ) 0.57 (components n<14 in the figure) RC (Assume that log 10 ( ( B n ) C ) does not depend on n.) ( ) RC = R E 0.5 R E 3300 km When RC R E, k 0 (components n>14 in the figure) 8

29 World Digital Anomaly Map (WDMAM 007). Worldwide distribution of anomalies in the magnetic lithosphere. 9

30 International Geomagnetic Reference Field (IGRF) 30

31 Geomagnetic field models are represented as spherical harmonic expansions of a scalar magnetic potential. Such a model can then be evaluated at any desired location to provide the magnetic field vector. IGRF was introduced by the International Association of Geomagnetism and Aeronomy (IAGA) in 1968 in response to the demand for a standard spherical harmonic representation of the Earth's main field. IGRF can be considered to consist of two parts: mathematical functions that describe how each multipole field changes as a function of latitude, longitude, and radius ( geometry of the multipole field ) coefficients (n+1 for each n 1) associated with each multipole ( strength of the multipole field ) 31

32 IGRF is meant to give a reasonable approximation, near and above the Earth's surface, to that part of the Earth's magnetic field which has its origin inside the surface. The model is updated at 5-year intervals. The latest (as of May 015) is IGRF-1. At any one epoch, the IGRF specifies the numerical coefficients of a truncated spherical harmonic series. For dates until 000 the truncation is at n = 10, with 10 coefficients. From 000 the truncation is at n = 13, with 195 coefficients. Such a model is specified every 5 years, for epochs , , etc. For dates between the model epochs, coefficient values are given by linear interpolation. For the 5 years after the most recent epoch there is a linear secular variation model for forward extrapolation; this SV model is truncated at n = 8, so has 80 coefficients (in effect the next 40 or 115 coefficients are defined to be zero). (t t 0 ) g (t )= g (t 0 )+ g ' (t 0 )(t t 0 )+ g ' ' (t 0 ) +! (t t 0 ) m m m m h n (t)=h n (t 0 )+h' n (t 0 )(t t 0 )+h ' ' n (t 0 ) +! m n m n m n m n 3

33 Health warning When using IGRF, to avoid ambiguity you should state explicitly which IGRF generation you are using. Because of the time variation of the field, really good models can only be produced for times when there is global coverage by satellites measuring the vector field. This occurred in: : MAGSAT 1999 : Ørsted, CHAMP, SWARM At some time later, IGRF models are replaced by definitive DGRF models ( definitive = we will not be able to do significantly better in the future ). Interpolate between the appropriate DGRF models if they exist. If there is not a DGRF model, then use the appropriate IGRF model. If you measure the magnetic field at a point on the Earth's surface, do not expect to get the value predicted by the IGRF: The numerical coefficients will not be correct: the model field produced will differ from the actual field. Because of truncation, the IGRF model represents only the lower spatial frequencies (longer wavelengths) of the field: higher spatial frequency components are not accounted for. There are also other contributions to the observed field (both natural and man-made) the IGRF is not trying to model: buildings, parked cars, magnetization of crustal rocks, traffic, DC electric trains and trams, electric currents in the ionosphere and magnetosphere, etc. 33

34 IGRF-1 Full name: IGRF 1th generation Short name: IGRF-1 Valid for: Definitive for: Reference: Thébault et al., Earth Planets and Space,

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36 Magnetic maps of the Earth from 36

37 Isogonic line (D = constant) Map of declination (degrees east or west of true north) at Agonic line (D = 0o) 37

38 Map of predicted annual rate of change of declination (degrees/year east or west) for

39 Map of inclination (angle in degrees up or down that magnetic field vector is from the horizontal) at Magnetic equator (I = 0o) 39

40 Map of predicted annual rate of change of inclination (degrees/year up or down) for

41 Minima around magnetic poles Map of horizontal intensity at Maximum around equator 41

42 Map of predicted annual rate of change of horizontal intensity for

43 Map of vertical intensity at

44 Map of predicted annual rate change of vertical intensity for

45 Maximum associated with north magnetic pole (note asymmetry between hemispheres) Map of total intensity at Maximum associated with south magnetic pole 45

46 Map of predicted annual rate of change of total intensity for

47 Average multipole strengths Average annual rate of change of multipole strengths 47

48 Maps of the dipole and non-dipole components of IGRF in Finland 48

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63 Altitude-Adjusted Corrected Geomagnetic Coordinates (AACGM) Examples of determining AACGM coordinates for four geographic locations along the prime meridian. Red lines represent IGRF field lines emanating from geographic starting locations at 50o, 40o, 30o latitude, and ending at the Earth-centered magnetic dipole equator. AACGM coordinates are given by the coordinates of the dipole field lines, shown in green. The magenta line shows the IGRF field line starting at 0 o latitude, which intersects the surface of Earth before the dipole equator. AACGM coordinates are undefined for such locations. The region near the magnetic dip equator (orange line) which includes these field lines is marked by yellow lines on Earth s surface. From: Shepherd (014). 63