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Aspects of geophysical potential theory

Joachim Vogt

Jacobs University Bremen

Course 210392Earth and Planetary Physics

Spring 2009

Joachim Vogt (Jacobs University Bremen) Aspects of geophysical potential theory Course 210392, Spring 2009 1 / 36

Motivation

The Earth’s gravitational field and the geomagnetic fields are potential

fields : they can be written as the gradient of a scalar potential Φ.

Which differential equation is satisfied by the potential?

What is a convenient mathematical representation (series expansion) ?

Geophysical fields are typically measured at the surface but we wish to

know their values also in other regions.What is the relationship between potential values in different regions?

Which information is required to determine the potential uniquely?

Geophysical fields are caused by source distributions. E.g., the Earth’sgravity field is generated by the mass density distribution in the interior.

How does the potential depend on the source distribution ?

What can surface measurements tell us about the sources?


Motivation (2)

Earth’s gravity anomaly at the surface based on GRACE satellite data

[(1) GRACE / GFZ Potsdam]


Motivation (3)

Gravity anomaly in the Earth’s mantle based on GRACE satellite data

[(1) GRACE / GFZ Potsdam]




Motivation (4)

Geomagnetic field models based on data from the CHAMP satellite

Field intensity at the surface

[(2) CHAMP / S. Maus / NOAA-NGDC]

Vertical field component at . . .

. . . the Earth’s surface

. . . the core-mantle boundary

[(3) CHAMP / GFZ Potsdam]


Motivation (5)

Lithospheric magnetic field model (MF6): anomaly vertical component

[(4) CHAMP / S. Maus / GFZ Potsdam / NOAA-NGDC]


Motivation (6)

Lithospheric magnetic field model (MF4): anomaly total intensity . . .

. . . at 400 km altitude ...at 50 km altitude

[(4) CHAMP / S. Maus / GFZ Potsdam / NOAA-NGDC]


Overview

Part I: Vector field representations and potential equations

Potential representations of vector fields

The equations of Poisson and Laplace

Part II: Spherical harmonics

Solution of the Laplace equation in spherical coordinates

Properties of spherical harmonics

Part III: Earth’s global potential fields

Earth’s gravity field, Stokes coefficients

Geomagnetic field, Gauß coefficients




Aspects of geophysical potential theory – Part I

Vector field representations

and potential equations


Curl-free vector fields and scalar potentials

A vector field V is called

curl-free if ∇× V = 0,

a potential field (or gradient field ) if a scalar function Φ exists suchthat V = −∇Φ. Then the function Φ is the potential of V .

Note that

potential fields are always curl-free because ∇×∇Φ = 0,

the reverse holds for suitable definition domains (make sure thatsources are not enclosed ),

thus the term curl-free field is used usually as a synonym for potential field .

Examples:

Gravitational potential Φg, gravitational acceleration g = −∇Φg.

Electrostatic potential Φe, electric field E = −∇Φe.


Divergence-free vector fields and vector potentials

A vector field B is called divergence-free if ∇ ·B = 0. Such a field is thecurl of a vector potential A (suitable definition domain assumed):

B = ∇×A .

The vector potential is not uniquely defined, and additional constraints arerequired: gauge conditions , e.g.,

Coulomb gauge (magnetostatics): ∇ ·A = 0,

Lorentz gauge (electrodynamics): ∇ ·A = −∂ Φe/∂t,

A ·B = 0 (nonlinear gauge condition)→ Euler potentials α, β : B = ∇α ×∇β .

Special representations: toroidal and poloidal fields (dynamo theory).


Examples of potential equations (1)

Electrostatics : inserting E = −∇Φe into Gauß’ law yields

e = ∇ ·D = ε0∇ ·E = ε0∇ · (−∇Φe) = −ε0∇2Φe

(ε0: permittivity of free space, e: charge density), and we obtain

∇2Φe = −e/ε0

which is Poisson’s equation.

Magnetostatics I (magnetic fields due to magnetization): ∇×H = 0(zero currents), thus H = ∇Φm. B = µ0(H +M ), ∇ ·B = 0, thus

∇2Φm = −∇ ·M

(B: magnetic flux density [Vs/m2], H : magnetic field [A/m],M : magnetization [(A/m2)/m3], µ0: vaccum permeability).




Examples of potential equations (2)

Magnetostatics II (magnetic fields produced by stationary currents):zero magnetization, ∇ ·B = 0, thus B = ∇×A. From ∇×B = µ0 j

(Ampere’s law) we obtain

µ0 j = ∇×B = ∇×∇×A = ∇(∇ ·A)−∇2A .

Combine with Coulomb gauge ∇ ·A = 0 to get

∇2A = −µ0 j , ∇ ·A ,

i.e., a system of three Poisson-type equations (plus gauge condition).

Gravitational potential and mass density :

∇2Φg = 4πG

(: mass density, G: gravitational constant).


The equations of Poisson and Laplace

The potentials satisfy Poisson-type equations

∇2Φ = σ

where

Φ is the potential,

σ is the source term, and

∇2 is the Laplace operator.

In regions without sources: σ = 0. Poisson’s equation reduces to

∇2Φ = 0

This is the Laplace equation.

Solutions of the Laplace equations are called harmonic functions .


Boundary value problems

Poisson’s equation and the Laplace equation are elliptic PDEs that haveunique solutions if suitable boundary conditions are given.

Boundary value problem: Find the solution Φ of ∇2Φ = σ in a volume V for given values of the source function σ = σ(r) in V and additionalinformation on the boundary surface S = ∂ V .

(D) Dirichlet problem: Given on S = ∂ V are the values of the potential Φ.

(N) Von Neumann problem: Given on S = ∂ V are the values of thederivative of Φ in the direction n normal to the boundary surface:

∂ Φ

∂n= n · ∇Φ .

Both (D) and (N) have unique solutions under reasonable conditions.


Aspects of geophysical potential theory – Part II

Spherical harmonics




Laplace equation in spherical coordinates

Solutions of the Laplace equation ∇2Φ = 0 in spherical coordinates(r,ϑ,λ) are called (solid) spherical harmonics :

∇2Φ =1

r2

∂

∂r

r2 ∂ Φ

∂r

+

1

r2 sin ϑ

∂

∂ϑ

sin ϑ

∂ Φ

∂ϑ

+

1

r2 sin2 ϑ

∂ 2Φ

∂λ2= 0 .

We seek solutions through separation of variables which yields threeordinary differential equations (ODEs): write the potential Φ as a product

Φ(r,ϑ,λ) = R(r) · Ψ(ϑ, λ)

and insert this ansatz into the Laplace equation. Here

R = R(r) is called a radial function, and

Ψ = Ψ(ϑ, λ) is a surface spherical harmonic .


Separation of variables (1)

After multiplication with r2/Φ, we obtain

r2∇2Φ

Φ=

1

R

d

dr

„r2

dR

dr

« | z =const=n(n+1)

+1

Ψsin ϑ

∂

∂ϑ

„sin ϑ

∂ Ψ

∂ϑ

«+

1

Ψsin2 ϑ

∂ 2Ψ

∂λ2 | z =−n(n+1)

= 0 .

Note that the first underbraced expression depends only on the radial coordinate r, and thesecond one only shows angular dependences. This implies that each expression must be aconstant which is written as ±n(n + 1) for later convenience.

The equation for the radial function R reads

d

dr

„r2

dR

dr

«= n(n + 1) R

which can be solved by the ansatz R ∝ rα to yield two independent solutions rn and r−(n+1).

General solution for R = R(r) (radial function of degree n):

R(r) = A rn + B r−(n+1) .



We now separate the variables further and write Ψ as the product

Ψ(ϑ, λ) = Θ (ϑ) · Λ(λ) .

The differential equation for the function Ψ reads

1

sin ϑ

∂

∂ϑ

„sin ϑ

∂ Ψ

∂ϑ

«+

1

sin2 ϑ

∂ 2Ψ

∂λ2+ n(n + 1)Ψ = 0 ,

We insert the separation ansatz into this PDE and multiply by sin2 ϑ/Ψ to obtain

sin ϑ

ϑ

d

dϑ

„sin ϑ

dϑ

dϑ

«+ n(n + 1)sin2 ϑ | z

=const=m2

+1

Λ

d2Λ

dλ2 | z =−m2

= 0 .

Since the first term depends only on ϑ and the second one only on λ, we can conclude as beforethat each expression must be a constant. The azimuthal function Λ(λ) satisfies

d2Λ

dλ2+ m2Λ = 0 .



For the azimuthal dependence, the general solution can be written as

Λ(λ) = C cos mλ + D sin mλ .

Note that due to the uniqueness requirements Λ(λ) = Λ (λ + 2π) and Λ(λ) = Λ(λ + 2π),the number m must be a real integer and one can further assume that m is non-negative.

The ODE for Θ = Θ(ϑ) is more complicated than the previous ones:

1

sin ϑ

d

dϑ

„sin ϑ

dΘ

dϑ

«+˘

n(n + 1)sin2 ϑ− m2¯

Θ = 0 .

Substituting µ = cos ϑ and M (µ) = Θ(ϑ) yields

d

dµ

(1 − µ2)

dM

dµ

+

n(n + 1)sin2 ϑ −

m2

1 − µ2

M = 0

This ODE is solved by Legendre functions P mn (µ) .




Legendre functions (1)

Terminology:

m = 0: Legendre polynomials P n(µ),

m = 0: associated Legendre functions P mn (µ).

Rodriguez’ formula for Legendre polynomials:

P n(µ) = 12nn! dn

dµn

µ2 − 1

n

Associated Legendre functions:

P mn (µ) = pmn · (1− µ2)m/2 dm

dµmP n(µ)

Here pmn denotes a normalization factor.



Schmidt normalization: p0n = 1 (thus P n ≡ P 0n ), and

pmn =

2

(n− m)!

(n + m)!for m = 0 .

This is standard in geophysics and geomagnetism, and will also be used in

this course.

Other normalization conventions are given further below.The first four Legendre polynomials are

P 0 = 1 , P 1 = µ = cos ϑ ,

P 2 =1

2(3µ2 − 1) =

1

2(3cos2 ϑ− 1) ,

P 3 =1

2(5µ3 − 3µ) .

Associated Legendre functions

P 11 = (1− µ2)1/2 = sin ϑ ,

P 12 =√

3(1− µ2)1/2µ =√

3sin ϑ cos ϑ ,

P 22 =

√ 3

2(1− µ2) =

√ 3

2sin2 ϑ .



(5) Wikipedia Commons



Further terminology and important properties

n is called degree , and m is called order (n ≥ 0, 0 ≤ m ≤ n),

P n(µ) is a polynomial of degree n with n zeroes in ] − 1, 1[,

Orthogonality: +1−1 P k(µ)P n(µ) dµ = 0 if n = k,

P mn (µ) has (n− m) zeroes in ] − 1, 1[,

P n(1) = 1 and P n(−1) = (−1)n

for all n = 0, 1, 2, . . .The generating function of the Legendre polynomials is

1 1 − 2µh + h2

=

∞n=0

hnP n(µ) , |h| < 1

Recursion formula:

(n + 1)P n+1 = (2n + 1)µP n − nP n−1




Surface spherical harmonics (1)

The elementary surface spherical harmonics are given by

Ψmσn (ϑ, λ) = P mn (cos ϑ) ·

cos mλ , σ = c

sin mλ , σ = s

for 0 ≤ m ≤ n. Note that Ψ0sn ≡ 0.

Every function f (ϑ, λ) can be represented as a convergent series of

elementary surface spherical harmonics, i.e.,

f (ϑ, λ) =

∞n=0

nm=0

σ=c,s

Amσn Ψmσ

n (ϑ, λ)

with the coefficients (Ω is the unit sphere, and dΩ = sin ϑ dϑ dλ):

Amσn =

2n + 1

4π

Ω

f (ϑ, λ) Ψmσn (ϑ, λ) dΩ .


Surface spherical harmonics (2)

The functions Ψmσn are called

zonal surface sphericalharmonics if m = 0,

sectorial surface sphericalharmonics if m = n,

tesseral surface sphericalharmonics if 0 = m = n.

In the figure, l corresponds to n.

The lines indicate the zeroes of the functions Ψmσ

n .

(5) Wikipedia Commons


Normalization conventions (1)

Normalization of (associated) Legendre functions: consider K n and K mn in

m = 0 : P n(µ) = K n · 1

2nn!

dn

dµn(µ2 − 1)n , P 0n ≡ P n ,

m = 0 : P mn (µ) = K mn · (1− µ2)m/2dm

dµmP n(µ) .

Here is a list of several popular normalization conventions.

(1) No specific normalization: K n = K mn = 1 (mathematics, physics, geodesy). In this casewe obtain Z +1

−1[P n(µ)]2 dµ =

Z +1−1

[P mn (µ)]2 dµ =2

2n + 1

(n + m)!

(n− m)!.

(2) “Full/complete normalization I”:

K n =

r 2n + 1

2, K mn =

r 2n + 1

2

s (n − m)!

(n + m)!.

This is useful if Legendre functions are to be normalized on the interval [−1, 1], and notas part of the surface spherical harmonics. One obtainsZ +1

−1[P n(µ)]2 dµ =

Z +1−1

[P mn (µ)]2 dµ = 1 .


Normalization conventions (2)

Normalization conventions for associated Legendre functions (continued)

(3) Schmidt normalization:

K n = 1 , K mn =

s 2

(n− m)!

(n + m)!.

This normalization yields

Z π

0

sin ϑ dϑZ 2π

0

dλ [P mn (cos ϑ)cos mλ]2 =4π

2n + 1

which is independent of m. No additional factor in the addition theorem (see below).

(4) “Full/complete normalization II”:

K n =√

2n + 1 , K mn =√

2n + 1

s 2

(n − m)!

(n + m)!.

The integral1

4π

Z π0

sin ϑ dϑ

Z 2π0

dλ [P mn (cos ϑ)cos mλ]2 = 1

is independent of m. This normalization is used, e.g., in satellite geodesy (Gaposhkin).




Further identities for Legendre functions

Addition theorem:

ϑ and λ: co-latitude and the azimuth of a point P ,

ϑ and λ: co-latitude and the azimuth of a point P ,

both points lie on the unit sphere, and χ is the angle between P and P , then

cos χ = cos ϑ cos ϑ + sin ϑ sin ϑ cos(λ− λ) .

For Legendre functions in Schmidt normalization:

P n(cos χ) =nX

m=0

P mn (cos ϑ)P mn (cos ϑ)cos m(λ− λ)

Orthogonality of elementary surface spherical harmonics (in Schmidt normalization):

Z Ω

Ψmσn Ψmσ

n dΩ =

8<:

4π2n+1 : n = n, n = n, σ = σ =

s, c : m = 0 ,

c : m = 00 : else .


Aspects of geophysical potential theory – Part III

Earth’s global

potential fields


Gravitational potential

Consider a mass density distribution that is completely enclosed in asphere of radius a. Thus = 0 for r > a, the gravitational potential Φsatisfies ∇2Φ = 0, and can be written in terms of spherical harmonics:

Φ(r,ϑ,λ) =∞n=0

nm=0

rn P mn

(cos ϑ) [Amn

cos(mλ) + Bmn

sin(mλ)]

+∞

n=0

n

m=0

r−(n+1) P mn

(cos ϑ) [Amn

cos(mλ) + Bmn

sin(mλ)]

The potential must remain finite as r →∞, thus Amn = Bm

n = 0.

Earth: a = equatorial radius, and the largest term is A00/r = −GM E/r

(compare with spherical Earth model), thus A00 = −GM E and

Φ(r) = −GM E

r

1 −

∞n=1

nm=0

a

r

n

P mn

(cos ϑ) [C mn

cos(mλ) + S mn

sin(mλ)]

The parameters C mn and S mn are called Stokes coefficients .Joachim Vogt (Jacobs University Bremen) Aspects of geophysical potential theory Course 210392, Spring 2009 31 / 36

Stokes coefficients and density distribution (1)

The Earth’s gravitational potential at a point P in the exterior can b e written as follows:

Φ(r) = −G

Z Earth

(r)

|r− r|d3r = −G

Z Earth

(P )

D(P, P )dV

Primed variables refer to source (mass) distribution. Vectors r and r point from the origin to

P and P , respectively.

The distance D(P, P ) between P and P is expressed through

D2 = r2 + r2 − 2rr cos χ

where χ = ∠(r,r). The reciprocal distance 1/D = 1/|r− r| can be written using Legendrepolynomials as follows

1

D=

1

|r− r| =

1

r

∞Xn=0

„r

r

«nP n(cos χ) , r ≥ r ,

1

D=

1

r

∞Xn=0

“ r

r

”lP n(cos χ) , r ≤ r .

With this expansion, the gravitational potential reads

Φ(r) = −G

r

Z Earth

(r)Xn

„r

r

«nP n(cos χ) d3r .




Stokes coefficients and density distribution (2)

Inserting the addition theorem for spherical harmonics

P n(cos χ) =nX

m=0

P mn (cos ϑ) P mn (cos ϑ) cos(m(λ − λ))

then yields

Φ(r) = −∞Xn=0

G

rn+1

nXm=0

P mn (cos ϑ)

Z Earth

(r)rnP mn (cos ϑ)cos(m(λ − λ)) d3r

where cos(m(λ

− λ)) = cos(mλ) cos(mλ

) + sin(mλ) sin(mλ

) . The term−(G/r)R

Earth (r) d3r = −GM E/r is separated from the expansion for normalizationpurposes, and the series can now be written in the form

Φ(r) = −GM Er

(1 −

∞Xn=1

nXm=0

“a

r

”nP mn (cos ϑ) [C mn cos(mλ) + S mn sin(mλ)]

)

if we relate the Stokes coefficients to the mass density distribution as follows:C mnS mn

ff=

−1

anM E

Z V

(r,ϑ,λ) rn P mn (cos ϑ)

cos mλsin mλ

ffdV .

Note that dV = r2 sin ϑ dr dϑ dλ, and P mn are the Schmidt normalized Legendre functions.


Scalar potential of the geomagnetic field

In geomagnetism, the term magnetic field is mostly used for the fieldB = µ0H . The field B is measured in Tesla [T], and in other contexts itis called magnetic induction or magnetic flux density.

In the neutral atmosphere (between the ground and the ionosphere), thereare neither electrical currents nor magnetization, thus ∇×B = 0 and

B = −∇Φ .

Here Φ is the scalar magnetic potential, and (because of ∇ ·B = 0)

∇2Φ = 0 .

How to separate internal and external contributions to the geomagneticfield? Write the potential as a sum of two terms:

Φ = Φi + Φe .


Gauß coefficients

Decomposition according to C. F. Gauß

Φi = RE

∞n=1

RE

r

n+1 nm=0

(gmn cos mλ + hm

n sin mλ) P mn (cos ϑ) ,

Φe = RE

∞

n=1

r

REn n

m=0

gmn cos mλ + hm

n sin mλ P mn (cos ϑ)

The parameters gmn , hm

n , gmn , hm

n are called Gauß coefficients . Note thatthey have the same physical dimension as B and are usually given in nT.

Historically, the Gauß coefficients were determined from surfacemeasurements of the magnetic elements X (north), Y (east), and Z

(down). Nowadays they are determined from space (missions MAGSAT,CHAMP, Ørsted) and only supplemented by ground or aeromagneticmeasurements (mainly for regional field models).


Figure references

(1) Preliminary gravity anomaly maps based on data from the GRACE mission, see the websites hosted by the Geoforschungszentrum (GFZ) Potsdam athttp://op.gfz-potsdam.de/grace/results/grav/g001 eigen-grace01s.html (10April 2009). See also the GRACE web page at the University of Texas(http://www.csr.utexas.edu/grace/ ).

(2) Magnetic field intensity map based on the model POMME 3.0 by Stefan Maus, see theweb site http://www.geomag.us/info/mainfield.html hosted by the CooperativeInstitute for Research in Environmental Sciences (CIRES) and NOAA’s National

Geophysical Data Center (NGDC). See also the web site http://geomag.org/ (10 April2009).

(3) Maps of the vertical field component based on recent geomagnetic field models. Imagecredit: Geoforschungszentrum (GFZ) Potsdam, see http://www.gfz-potsdam.de/ (10April 2009).

(4) Lithospheric magnetic field maps based on the Magnetic Field Models MF4 and MF6developed at the Geoforschungszentrum (GFZ) Potsdam (http://www.gfz-potsdam.de/ )and at NOAA’s National Geophysical Data Center (NGDC). See the MF6 web site athttp://www.geomag.us/models/MF6.html by Stefan Maus at the NGDC (10 April 2009).

(5) Image files Harmoniques spheriques positif negatif.png and Legendre poly.svg fromWikipedia Commons http://en.wikipedia.org/wiki/ (10 April 2009).


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