9. Spherical Harmonics

1 ORTHOGONAL FUNCTIONS249. Spherical HarmonicsNow we come to some of the most ubiquitous functions in geophysics,used in gravity,geomagnetism and Harmonics are the Fourier series for functions can are used to build solutions to Laplace sequation andother differential equations in a Spherical shall treat Spherical Harmonics as eigensolutions of the surface would be like developing Fourier series as eigensolutions of the operator (d/dx)2on a finite line,but with boundary conditions thatyanddy/dxmatchatthe get some mileage from representing a thing in two ways, onewithin a fixed coordinate system, the other in coordinate-free we need aspherical polar coordinate system: see the originOis alwaysfixed to bethe center of the unit sphere,and all coordinates are referred to that usdefine asurface gradientfor the sphere in two ways: 1= + sin (1)=r r r.

2 (2)The subscript one is to remind us the operator acts over the unit sphere,S(1). Thefirst 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 justsubtracts out the radial part. The second definition shows the operator is indepen-dent of coordinate orientation and also that nothing funny happens at the ordinary Laplacian operator in IR3is 2= xi xi= 2 x2+ 2 y2+ 2 z2(3)=1r 2 r2r+1r2 21(4)where 21is thesurface Laplacian,sometimes also called theBeltrami operator;Figure 9: The Spherical polar coordinate systemORTHOGONAL FUNCTIONS25relative to a coordinate system 21= 2 2+cot +1sin2 2 2=1sin sin +1sin2 2 2.

3 (5)We can think of 21as the ordinary Laplacian, with the radial part subtracted andscaled byr2to make it unitless: from (4) 21=r2 2 r 2 r2r.(6)As with (2), this equation shows the operator is independent of any coordinate sys-tem and the singularity at the poles is not intrinsic to 21,but is an artifact of there is another surface operator s=r 1 1;this one hasdimensions of 1/Lliked/dxor the regular gradient operator .To develop Spherical Harmonics we ask for the eigenvalues and eigenfunctionsof the surface dothis in part because,just as in IR3the eigenvectorsof a symmetric matrix provide an orthogonal basis for the space,sohere the selfadjoint operator has a collection of orthogonal functions that span the complex-valued functions on the unit sphere as elements in theHilbert space complexL2(S(1)); hereS(1) is shorthand for the unit sphere,the set ofpoints |r|=1.

4 Thenthe norm is||f||= S(1) d2 s|f( s)|2 (7)and the notationd2 sis a surface element onS(1), whichwould be sin d d in aparticular polar coordinate find it handy to work complex functions onS(1). ThespaceL2(S(1)) comes with the inner product(f,g)=S(1) d2 sf( s)g( s) (8)It can be show that the service Laplacian is self adjoint, that is( 21f,g)=(f, 21g). (10)Furthermore,aswehavealready remarked, the eigenfunctions of self adjoint opera-tors are ( r)=u( , )satisfies the eigenvalue equation 21u= u(11)anduis continuous everywhere onS(1) but is not identically zero,thenuis aneigenfunction for every value of can support when is one of the integers =0, 2, 6, 12, 20.

5 L(l+1)..(12)does (11) have nontrivial does one prove this?The traditional way isby a technique calledseparation of variables,assuming thatucan be written as aproduct of two single-argument functions:u( , )= ( ) ( ), substituting in andORTHOGONAL FUNCTIONS26getting two one-dimensional eigenvalue problems,one eachfor and .See Morseand Feshbach,Methods of Mathematical Physics,Vol II, 1953, for ,Parker and Constable (Foundations of Geomagnetism,1996) does thisentirely differently,bylooking at homogeneous harmonic calllthedegreeof the Spherical eigenfunctions of 21asso-ciated with the eigenvalues are calledspherical Harmonics ;wewrite themu( , )=Yml( , ).

6 (13)We will give an explicit formula for these functions later; they are complex-valued onthe eigenvalues in (12) are not simple (except =0), but 2l+ means that associated with the eigenvalue = 6= 2 (2+1),say, there are 5=2 2+1different (that is,linearly independent) is where the indexmcomes in: for eachdegreeltheordermcan be any one of l, l+1,..0, 1,..l,giving the required number of different functions for traditional arrangement these are chosen to be mutually orthogonal.

7 Thus thespherical Harmonics form an orthogonal family:(Yml,Ykn)=S(1) d2 sYml( s)Ykn( s) =0,m korl n.(14)We usually scale the Spherical Harmonics to be of unit norm:||Yml||=1(15)then the Spherical Harmonics are said to befully normalized,although not every-one does fully normalized Harmonics (14) and (15) combine to give(Yml,Ykn)= ln mk.(16)Notice that any linear combination of eigenfunctions of degreelis also a valid eigen-function with eigenvalue l(l+1).It is time to write out an explicit form solutions are the onesobtained by the separation of variables mentioned earlier they are eachaproductof a function of (colatitude) and one of (longitude).

8 Herewe go:Yml( , )=Nlmeim Pml(cos )(17)whereNlmis a normalization constant to adjust the size of the functions; as I men-tioned earlier,Iusually choose it to enforce (15); thenNlm=( 1)m 2l+14 (l m)!(l+m)! .(18)There are however several different conventions example,leav-ing off the alternating sign and removing the factor ((2l+1)/4 ) ,results in func-tions that areSchmidt convention, (18), is most convenient for the-oretical work, because of (15), whichthe others fail to comply factorexp(im )isjust the complex Fourier basis for functions of longitude on complexL2( , ).

9 Manyold-fashioned authors (in geomagnetism especially) use sines andcosines with real coefficients here last factor is called anAssociatedLegendre functionand is defined byORTHOGONAL FUNCTIONS27 Pml( )=12ll!(1 2)m/2 l+m l+m( 2 1)l.(19)When the orderm=0, the Associated Legendre function becomes a polynomial in and instead being writtenP0l( )itisdesignatedPl( ), theLegendre polynomialwhichhaveseen several time a very short =(1 2) .P0( )=1P1( )= P11( )=sP2( )=(3 2 1)/2P12( )=3 sP22( )=3s2P3( )= (5 2 3)/2P13( )=3s(5 2 1)/2P23( )=15s2 P33( )=15s3P4( )=(35 4 30 2+3)/8P14( )=5s (7 2 3)/2P24( )=15s2(7 2 1)/2P34( )=105s3 P44( )=105s4P5( )= (63 4 70 2+15)/8P15( )=15s(21 4 14 2+1)/8P25( )=105s2 (3 2 1)/2P35( )=105s3(9 2 1)/2P45( )=945s4 P55( )= havelisted only positivemsince there is a nice symmetry that allows us to dowithout:Y ml=( 1)m(Yml).

10 (20)We also note these special values they are worth remembering:Pl(1)=1,Pml(1)=0,m 0,Pll(cos )=clsinl .(21)Next we note one of the most important properties of the Spherical Harmonics :they are acompleteset for expanding functions onL2(S(1)). Thusin the usual way,whenf( , )= l=0 lm= l clmYml( , )(22)we get the coefficients fromORTHOGONAL FUNCTIONS28clm=(f,Yml)=S(1) d2 sf( s)Yml( s) .(23)It is this property that makes Spherical Harmonics so useful. Orthogonality is aproperty that follows from the self-adjointness of follows from amore subtle property,that the inverse operator of 21is compact, a property thatwould take us too far afield to you have seen them before,you probably have no idea what the spheri-cal Harmonics look like at this point.