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9. Spherical Harmonics Now we come to some of the most ubiquitous functions in geophysics,used in gravity, geomagnetism and seismology.Spherical harmonics are the for the .These functions can are used to build solutions to Laplace’sequation and other differential equations in a spherical setting. We shall treat spherical harmonics as eigensolutions of the surface Laplacian. This would be like developing Fourier series as eigensolutions of the operator (d/dx)2 on a finite line,but with boundary conditions that y and dy/dx matchatthe two ends.Wesometimes get some mileage from representing a thing in two ways, one within a fixed coordinate system, the other in coordinate-free form. First we need a spherical polar coordinate system: see the figure.The origin O is alwaysfixed to be the center of the unit sphere,and all coordinates are referred to that origin. Let us define a surface gradient for the sphere in two ways: ∂ φˆ ∂ ∇ = θˆ + (1) 1 ∂θ sin θ ∂φ

∂ = r∇−r .(2) ∂r The subscript one is to remind us the operator acts over the unit sphere, S(1). The first definition shows how to compute the surface gradient in a spherical polar coor- dinate system; the second assumes the function is defined in all of space and just subtracts out the radial part. The second definition shows the operator is indepen- dent of coordinate orientation and also that nothing funny happens at the poles. The ordinary Laplacian operator in IR3 is ∂ ∂ ∂2 ∂2 ∂2 ∇2 = = + + (3) 2 2 2 ∂xi ∂xi ∂x ∂ y ∂z

1 ∂2 1 = r + ∇2 (4) r ∂r2 r2 1 ∇2 where 1 is the surface Laplacian,sometimes also called the Beltrami operator;

Figure 9: The spherical polar coordinate system ORTHOGONAL FUNCTIONS 25 relative to a coordinate system ∂2 ∂ 1 ∂2 1 ∂ ∂ 1 ∂2 ∇2 = + θ + = θ + 1 cot sin .(5) ∂θ 2 ∂θ sin2 θ ∂φ 2 sin θ ∂θ ∂θ sin2 θ ∂φ 2 ∇2 We can think of 1 as the ordinary Laplacian, with the radial part subtracted and scaled by r2 to make it unitless: from (4) ∂2 ∇2 = r2∇2 − r r .(6) 1 ∂r2 As with (2), this equation shows the operator is independent of any coordinate sys- ∇2 tem and the singularity at the poles is not intrinsic to 1,but is an artifact of the −1 coordinates.Weremark there is another surface operator ∇s = r ∇1;this one has dimensions of 1/L like d/dx or the regular gradient operator ∇. To develop spherical harmonics we ask for the eigenvalues and of the surface Laplacian. We dothis in part because,just as in IR3 the eigenvectors of a symmetric matrix provide an orthogonal basis for the space,sohere the self adjoint operator has a collection of orthogonal functions that span the function space.Weregard complex-valued functions on the unit sphere as elements in the

Hilbert space complex L2(S(1)); here S(1) is shorthand for the unit sphere,the set of points |r| = 1. Then the norm is

 ½ || f || =  ∫ d2sˆ |f (sˆ)|2 (7) S(1)  and the notation d2sˆ is a surface element on S(1), whichwould be sin θ dθ dφ in a particular polar coordinate system. We find it handy to work complex functions on

S(1). The space L2(S(1)) comes with the inner product (f , g) = ∫ d2sˆ f (sˆ) g(sˆ)∗ (8) S(1) It can be show that the service Laplacian is self adjoint, that is ∇2 = ∇2 ( 1 f , g) (f , 1 g). (10) Furthermore,aswehavealready remarked, the eigenfunctions of self adjoint opera- tors are orthogonal. Suppose u = u(rˆ) = u(θ , φ)satisfies the eigenvalue equation ∇2 = λ 1u u (11) and u is continuous everywhere on S(1) but is not identically zero,then u is an ∇2 λ λ for 1.Not every value of can support suchsolutions.Only when is one of the integers λ = 0, − 2, − 6, − 12, − 20, ...− l(l + 1) ... (12) does (11) have nontrivial solutions.How does one prove this? The traditional way is by a technique called ,assuming that u can be written as a product of two single-argument functions: u(θ , φ) =Θ(θ ) Φ(φ), substituting in and ORTHOGONAL FUNCTIONS 26 getting two one-dimensional eigenvalue problems,one eachfor Θ and Φ.See Morse and Feshbach, Methods of Mathematical Physics,Vol II, 1953, for example.Chapter 3ofBackus,Parker and Constable (Foundations of Geomagnetism,1996) does this entirely differently,bylooking at homogeneous harmonic polynomials. ∇2 We call l the degree of the spherical harmonic.The eigenfunctions of 1 asso- ciated with the eigenvalues are called spherical harmonics;wewrite them θ φ = m θ φ u( , ) Y l ( , ). (13) We will give an explicit formula for these functions later; they are complex-valued on the sphere.The eigenvalues in (12) are not simple (except λ = 0), but 2l + 1-fold degenerate.This means that associated with the eigenvalue λ =−6 =−2 × (2 + 1), say, there are 5 = 2 × 2 + 1different (that is,linearly independent) eigenfunctions. This is where the index m comes in: for eachdegree l the order m can be any one of −l, − l + 1, ... 0, 1, ...l,giving the required number of different functions for any l.In the traditional arrangement these are chosen to be mutually orthogonal. Thus the spherical harmonics form an orthogonal family:

m k = 2 m k ∗ = ≠ ≠ (Y l , Y n) ∫ d sˆ Y l (sˆ) Y n(sˆ) 0, m k or l n .(14) S(1) We usually scale the spherical harmonics to be of unit norm: m = || Y l || 1(15) then the spherical harmonics are said to be fully normalized,although not every- one does this.With fully normalized harmonics (14) and (15) combine to give

m k = δ δ (Y l , Y n) ln mk .(16) Notice that any of eigenfunctions of degree l is also a valid eigen- function with eigenvalue −l(l + 1). m It is time to write out an explicit form for Y l .These solutions are the ones obtained by the separation of variables mentioned earlier − they are eachaproduct of a function of θ (colatitude) and one of φ (). Here we go: m θ φ = imφ m θ Y l ( , ) N lm e Pl (cos )(17) where N lm is a normalization constant to adjust the size of the functions; as I men- tioned earlier,Iusually choose it to enforce (15); then

 2l + 1½  (l − m)!½ N = (−1)m .(18) lm  4π   (l + m)!

There are however several different conventions regarding N lm.For example,leav- ing off the alternating sign and removing the factor ((2l + 1)/4π )½,results in func- tions that are Schmidt normalized.Our convention, (18), is most convenient for the- oretical work, because of (15), whichthe others fail to comply with. The factor exp(imφ)isjust the complex Fourier basis for functions of longitude on complex

L2(−π , π ). Many old-fashioned authors (in geomagnetism especially) use sines and cosines with real coefficients here instead. The last factor is called an Associated and is defined by ORTHOGONAL FUNCTIONS 27

1 ∂l+m Pm(µ) = (1 − µ2)m/2 (µ2 − 1)l .(19) l 2l l! ∂µ l+m When the order m = 0, the Associated Legendre function becomes a polynomial in µ 0 µ µ and instead being written Pl ( )itisdesignated Pl( ), the Legendre polynomial whichhaveseen several time already.Weprovide a very short table.Here s = sin θ = (1 − µ2)½.

P0(µ) = 1 µ = µ 1 µ = P1( ) P1( ) s µ = µ2 − 1 µ = µ 2 µ = 2 P2( ) (3 1)/2 P2( ) 3 sP2( ) 3s 2 P3(µ) = µ(5µ − 3)/2 1 µ = µ2 − P3( ) 3s(5 1)/2 2 µ = 2 µ 3 µ = 3 P3( ) 15s P3( ) 15s 4 2 P4(µ) = (35µ − 30µ + 3)/8 1 µ = µ µ2 − P4( ) 5s (7 3)/2 2 µ = 2 µ2 − P4( ) 15s (7 1)/2 3 µ = 3 µ 4 µ = 4 P4( ) 105s P4( ) 105s 4 2 P5(µ) = µ(63µ − 70µ + 15)/8 1 µ = µ4 − µ2 + P5( ) 15s(21 14 1)/8 2 µ = 2 µ µ2 − P5( ) 105s (3 1)/2 3 µ = 3 µ2 − P5( ) 105s (9 1)/2 4 µ = 4 µ 5 µ = 5 P5( ) 945s P5( ) 945s . We havelisted only positive m since there is a nice symmetry that allows us to do without: −m = − m m ∗ Y l ( 1) (Y l ) .(20) We also note these special values − they are worth remembering:

= m = ≠ l θ = l θ Pl(1) 1, Pl (1) 0, m 0, Pl(cos ) cl sin .(21)

Next we note one of the most important properties of the spherical harmonics: they are a complete set for expanding functions on L2(S(1)). Thus in the usual way, when

∞ l θ φ = m θ φ f ( , ) Σ Σ clm Y l ( , )(22) l=0 m=−l we get the coefficients from ORTHOGONAL FUNCTIONS 28

= m = 2 m ∗ clm (f , Y l ) ∫ d sˆ f (sˆ) Y l (sˆ) .(23) S(1) It is this property that makes spherical harmonics so useful. is a ∇2 property that follows from the self-adjointness of 1.Completeness follows from a ∇2 more subtle property,that the inverse operator of 1 is compact, a property that would take us too far afield to explore. Unless you have seen them before,you probably have no idea what the spheri- cal harmonics look like at this point. One further property of the Associated Legen- − µ m µ − dre functions helps: for 1< <1 the function Pl ( )crosses zero exactly l m times. m With this information we can picture the Y l on a sphere,bygraphing the places m m where Re Y l is zero; Im Y l has the same pattern, but rotated about the zˆ axis,as m can be seen at once from (17). So here is the recipe: for Y l there are two sets of m lines on the sphere where Re Y l vanishes: (a) a set of 2m equally-spaced halves of great circles through the poles (meridians) coming from the exponential; (b) a set of l − m small circles with planes normal to the z coordinate axis.See the pictures in Section 7. Forexample,when m = 0wealwayshavecircular symmetry about the z axis. After we have drawn a few we soon get the idea that the higher degree har- monics are shorter wavelength than the lower ones.Infact one can consider the sur- face harmonics to be standing waves in the surface of the sphere. Jeans’ formula says on a unit sphere 2π λ(l) = (24) l + ½ where λ(l)isthe wavelength of the waves of degree l.Notice this is independent of m and the position on the sphere.Inanalogy with Fourier filtering,people often make a spherical harmonic expansion, then remove the high-degree terms in order to emphasize the long-wavelength features; or conversely,the low degree-terms are removed to show the high-wavenumber behavior,ofthe , for example.Parse- val’sTheorem for a spherical harmonic expansion (22) gives

∞  l  2 2 || f || = Σ  Σ |clm|  (25) l=0 m=−l  The terms in square brackets gives a power spectrum of f as a function of recipro- cal wavelength or wavenumber (l is sometimes call the spherical wavenumber). This is a powerful way ofshowing how a geophysical field over a sphere is distributed according to its length scales. Two powerful properties of the spherical harmonics that do not follow immedi- ∇2 ately from their connection with 1 are the Addition Theorem and the generating function for .Westate the generating function first: 1 ∞ = Σ xl P (µ). (26) − µ + 2 ½ l (1 2 x x ) l=0 We can prove this later from the development of the potential of a point mass. ORTHOGONAL FUNCTIONS 29

The Spherical Harmonic Addition Theorem says 4π l P ˆ ⋅ ˆ = Y m ˆ Y m ˆ ∗ l(u v) + Σ l (u) l (v) .(27) 2l 1 m=−l The following proof won’t be found in any of the books; for another see Foundations of Geomagnetism,Chapter 3. Consider a delta function on S(1) whichpeaks at the point uˆ whichwewill write as δ (rˆ − uˆ ). From our discussion of the Reproducing Ker- nel (Section 8) when N → ∞ the function K N becomes a delta function. Taking the m complex nature of Y l into account, we find that (8.1) gives us the SH expansion of the delta function at uˆ :

∞ l δ − = m m ∗ (sˆ uˆ ) Σ Σ Y l (sˆ) Y l (uˆ ) .(28) l=0 m=−l Next consider this expansion for uˆ = zˆ,the north pole.The function is symmetric about the z axis,soall the coefficients in (28), vanish except those with m = 0. The same result follows from the second member of (21) whichimplies that m = m π φ = imφ m = ≠ Y l (zˆ) Y l (½ , ) N lme Pl (1) 0when m 0. Thus ∞ ∞ 2l + 1 δ (sˆ − zˆ) = Σ Y 0(sˆ)Y 0(zˆ)∗ = Σ P (cos θ )(30) l l π l l=0 l=0 4

∞ 2l + 1 = Σ P (zˆ ⋅ sˆ). (31) π l l=0 4 Next consider the delta function peak to be moved to some other point on the unit sphere,say uˆ ;then because (31) does not depend on the fact that zˆ is a coordinate axis,wecan replace zˆ by uˆ and the equation is still true: ∞ 2l + 1 δ (sˆ − uˆ ) = Σ P (uˆ ⋅ sˆ). (32) π l l=0 4 Equations (28) and (31) are expansions of the same function. We can compare these two expressions degree by degree.Since these are both expansions in the eigenfunc- ∇2 tions of 1 and the eigenfunctions of different degrees are orthogonal to eachother, the degree-l terms in eachsum must be equal to eachother: hence (27). Here are a couple of useful spherical harmonic expansions.Wecombine the generating function and the Addition Theorem together,tofind an expansion for the potential of point mass,not at the origin. With s < r we write 1 1 1 = = (33) R |r − s| (r2 + s2 − 2rs rˆ ⋅ sˆ)½

1 1 = .(34) r [1 + (s/r)2 − 2(s/r) rˆ ⋅ sˆ]½ Now we recognize the generating function (26):

∞  l 1 = 1 s ⋅ Σ   Pl(rˆ sˆ)(35) R r l=0 r ORTHOGONAL FUNCTIONS 30

∞ l 4π sl = Σ Σ Y m(rˆ) Y m(sˆ)∗ .(36) + l+1 l l l=0 m=−l 2l 1 r The last line follows from the Addition Theorem. So if we fix r = |r|and s and imagine moving around on the sphere of radius r,(36) is the spherical harmonic expansion of the potential 1/R as seen on that sphere. Asimilar expansion (whichwewon’t derive) used in scattering theory is the following.Suppose a plane wavecomes along and hits a sphere.The wavefield over the sphere can be decomposed in a spherical harmonic expansion: ∞ l J + (2π kr) e2π ik ⋅ r = (2π )3/2 Σ Σ eiπ l/2 l ½ Y m(rˆ) Y m(kˆ )∗ (37) π ½ l l l=0 m=−l (2 kr) where J l+½ is a Bessel function,animportant kind of function we will discuss later on in Fourier transforms. We mention the fact that the surface gradients of spherical harmonics are also orthogonal and complete in a certain sense.They can be used to expand surface vec- tor fields,that is vectors defined in S(1) that are alwaystangent to the sphere.The relevant relation is

2 ∇ m ⋅∇ k ∗ = + δ δ ∫ d sˆ 1Y l (sˆ) 1Y n(sˆ) l(l 1) ln mk (38) S(1) whichwecan interpret an an inner product on a different kind of on S(1). We mention briefly two more advanced topics.First, the 3- j symbols.Occa- sionally one runs into the need to perform integrals over the sphere of products of three spherical harmonics.These can alwaysbedone simply,and there are symme- try relationships that make many suchproducts vanish. The problem was first stud- ied systematically in of angular momentum, and so the refer- ence everyone uses is: Edmonds, Angular Momentum in Quantum Mechanics.There is a table made worthless because if its lackofsupporting explanation in Chap 27 Abramowitz and Stegun, Handbook of Mathematical Functions,Dover,1970. This book is worth knowing about in general, and has many results on Associated Legen- dre functions in Chaps 8 and 22. And second, one mayrequire a given spherical harmonic expansion in a rotated m coordinate system; how does the spherical harmonic Y l (r)transform to a different m set of coordinate axis? Obviously (perhaps) the new Y l will be a linear combination of harmonics of the same degree,asum over m only.But what are the coefficients? The Addition Theorem can be viewed as a special case.The general answer is com- plicated. See Winch’sarticle in in the Encyclopedia of Geophysics for this and much more. ORTHOGONAL FUNCTIONS 31

ATable of Spherical Harmonic Lore Property Formula Comments

1. Laplacian in polar 1 1 ∂2r ∇2 is angular part of ∇2 = ∇2 + 1 coordinates r2 1 r ∂r2 familiar Laplacian ∇2 m =− + m + 2. Eigenvalue 1 Y l l(l 1) Y l , There are 2l 1lin- l = 0, 1, 2, ... early independent eigenfunctions per l

3. Orthogonality 2 m k ∗ = True for every normal- ∫ d sˆ Y l (sˆ) Y n(sˆ) 0, ization unless l = n and m = k

Theoretician’snor- 2 m 2 = Other choices: 4π or 4. ∫ d sˆ |Y l (sˆ)| 1 malization 4π /(2l + 1) 5. Completeness ∞ l Works for any reason- = m f (sˆ) Σ Σ clmY l (sˆ) l=0 m=−l able function f on S(1) 6. Expansion coeffi- = 2 m ∗ Requires property 4 clm ∫ d sˆ f (sˆ) Y l (sˆ) cients 7. Addition Theorem 2l + 1 Requires property 4 P (sˆ ⋅ rˆ) = 4π l l m m ∗ Σ Y l (sˆ) Y l (rˆ) m=−l m π 8. Wavelength of Y l 2 Depends only on l + ½ degree l,not on order m or sˆ

m m 9. Appearance Re Y l vanishes on 2m Im Y l the same but meridians and l − m rotated about zˆ parallels

10. Parseval’sTheorem ∫ d2sˆ |f (sˆ)|2 = Requires property 4. ∞ l Get RMS value of f by 2 π Σ Σ |clm| dividing by 4 and tak- l=0 m=−l ing square root

11. Generating function 1 = Often used in conjunc- (1 − 2µr + r2)½ tion with property 7 ∞ l Σ r Pl(µ) l=0 12. Another orthogonal- 2 ∇ m ⋅∇ k ∗ = Requires property 4. ∫ d sˆ 1Y l (sˆ) 1Y n(sˆ) ity l(l + 1) δ ln δ mk ORTHOGONAL FUNCTIONS 32

10. Application: the Solution of Laplace’sEquation This is the application of spherical harmonics that we need for .We consider the region inside a spherical shell R1 ≤ r ≤ R2 inside whichLaplace’sequa- tion is obeyed by V,apotential: ∇2V = 0(1)

Sometimes we will let R2 tend to infinity,but it is useful to keep it finite for now. Let us write V as a function of spherical polar coordinates V(r, θ , φ)relative to O,the origin at the center of the concentric .Provided V is reasonably well behaved we can write it thus:

∞ l θ φ = m θ φ V(r, , ) Σ Σ V lm(r) Y l ( , ). (2) l=0 m=−l This follows because in any spherical surface the function V can be expanded in sur- face harmonics,and (2) simply says there is a different set of expansion coefficient for eachradius r.Anexpansion like (2) is perfectly general, and does not depend on V being harmonic.Let us now add the condition that V obeys Laplace’sequation, whichwewrite using (9.4): 1 ∂2rV 1 + ∇2V = 0. (3) r ∂r2 r2 1 Inserting (2) into (3) and recalling the eigenfunction property that ∇2 m =− + m 1Y l l(l 1) Y l (4) we obtain

∞ l 1 d2rV l(l + 1)   lm − V  Y m(θ , φ) = 0. (5) Σ Σ 2 2 lm l l=0 m=−l  r dr r  Because of the orthogonality of the surface harmonics the only way that this sum can be zero is if the factor in square brackets vanishes identically for every r and each l and m separately.Thus we find the ordinary differential equations: d2rV l(l + 1) lm − V (r) = 0. (6) dr2 r lm Notice the 2l + 1different functions with the same l but differing m obey the same differential equation. The standard way ofsolving suchequations is by substituting apower series.Omitting the details,weget B V (r) = Arl + (7) lm rl+1 as the most general solution. Clearly eachdegree and order in (5) will generally be associated with different constants A and B;thus the general solution to (3) is given by ∞ l  B  V(r, θ , φ) = Σ Σ A rl + lm Y m(θ , φ)(8)  lm l+1  l l=0 m=−l r ORTHOGONAL FUNCTIONS 33

where the constants Alm and Blm are coefficients in the spherical harmonic expan- sion of the potential; they must be determined by experiment or analysis for each different potential V. There is a physical interpretation of the two kinds of coefficients.Wesee that the contribution to V from the Alm grows with increasing radius r.This implies that the sources for Alm,that is the matter in the gravitational case,lie in the region outside R2,since V increases with proximity to the sources.Similarly Blm is asso- ciated with matter inside the inner sphere,radius R1.The sum of the Alm terms is called the external part of V and the Blm sum the internal part.Ifthe expansion (8) describes the gravitational field of the Earth, with O at its center,there can be no exterior part, as the is matter in question all lies within S(R1)Then it is conven- tional to rewrite (8) thus

Gm  ∞ l  al  =− E  + m θ φ  V 1 Σ Σ clm  Y l ( , ) .(9) r  l=2 m=−l r 

The numbers clm are sometimes called Stokes coefficients.Notice the absence of l = 1terms because O is at the center of mass so there is no term. Also note how the clm are dimensionless,asthe radial terms are scaled by the so-called refer- ence radius a.Aswementioned earlier, a is often the equatorial radius,because then the inner sphere encloses all the matter; sometimes the Earth’smean radius is used instead (often people forget to say!). Of course,practical models of the Earth’s potential do not sum to infinity,but very large degree model have been found with lmax = 360; how many coefficients is that?

Books Abramowitz, M., and I. A. Stegun, Handbook of Mathematical Functions,National Bureau of Standards,Dover Publications,New York, 1965. An essential reference work. Chapters on orthogonal polynomials,Bessel functions, numerical methods,etc.

Backus,G., Parker,R., and Constable,C., Foundations of Geomagnetism,Cambridge Univ Press,New York, NY,1996. Alot of material on spherical harmonics,including how to compute them.

Edmonds, Angular Momentum in Quantum Mechanics,Princeton, 1960. Astandard reference for 3- j symbols.

Morse,P.M.and H. Feshbach, Methods of Theoretical Physics, McGraw-Hill, New York, 1953. The bible of mathematical physics of 50 years ago.Still contains material not easily found elsewhere.

Winch, D., Spherical harmonics,in Encyclopedia of Geomagnetism and Paleomag- netism,Eds Gubbins D.and Herrero-Bervera, E., Springer,2007. Detailed treatment of spherical harmonics including transformations to rotated axes.