Spherical Harmonics - The Math Behind Solar Oscillations

 

A computer-generated image of some of the 10 million modes of acoustic waves on the Sun (red indicating receding wave fronts, and blue approaching).

The phenomenon of solar oscillations and their measurement & interpretation  -  being a relatively new field -  mainly developed in the last forty-five years. However, few lay folk are aware of it.  This is given the branch of mathematics - spherical harmonics-  is usually not taught until graduate level at most universities.

The first reckoning of non-radial solar oscillations arrived ca. 1968 with the work of Frazier who made two-dimensional plots of wavenumber vs. frequency or (k -w) diagrams. Several peaks of amplitude were found and it was suggested that these corresponded to the fundamental and first overtones for the solar envelope. Interestingly the patterns of solar oscillations - namely the acoustic or "p-modes" resemble those detected on drum heads by computer holography.

The Sun is clearly not a drum head, but it seems to behave like one in terms of its oscillations.  Solar physicists are particularly interested in what are called p, g and f modes given they are resonant modes of oscillation.  The p-modes are basically for acoustic or sound waves, the g modes are for internal gravity waves and the f modes are for surface gravity waves.   The spherical harmonic function, also peculiar to atomic physics, but in this post our focus is on the p-modes in solar oscillations context, given by:

y _nℓm  =  R _n (r)  Y _ℓm (q,j) exp (iw t)

Where R _n (r)   is applicable to all radial patterns with n the radial quantum number.

And for normalized spherical harmonics:

Y _ℓm (q,j) = 

(1)^m [2 ℓ +1 (ℓ - m)!/ 4n (ℓ +m)!] ^½ P ^ℓ_m (m  ) exp (i m j),

Where m  =  cos q and  w = 2 pn  where n = n _nℓm is the frequency of oscillation of mode n, ℓ, m. (Note: j is measured from the meridian of the ascending node of the Sun’s equator.)

The spherical harmonic, e.g.

 Y _ℓm (q,j),

determines the angular dependence of the eigenfunctions and hence the surface distribution of the oscillation amplitudes, i.e. as seen by an observer.

 The letters n, m and ℓ denote numbers whose meanings should be further clarified. The first is the radial order or the number of nodes in the radial direction. The second is the harmonic degree or azimuthal order which indicates the number of nodes around the equator on the three dimensional spherical surface. Finally we have the angular degree or the number of nodes from pole to pole, e.g. along longitude or meridian lines. The difference  (ℓ  - m) is also of interest as it yields the lines corresponding to parallels of latitude

Hence, there exist ℓ  values of  q  for which the function  P  _ℓ  (m  )  vanishes. The zeros occur on specific  parallels of latitude on the sphere. All odd-numbered harmonics vanish at the equator given they contain the factor  m  =  cos q.  Hence at the equator q    = 90 degrees so cos (90) = 0.  In like manner,

P m _ℓ   (m) vanishes along (ℓ  - m) parallels of latitude.  The associated functions vanish at the poles (m = +1) when m > 0.   The zeros at the poles are of order m/2 because of the factor :          (1 -  m ² ) ^m/2  in the general equation.[1]

The first few zonal harmonics are computed  using an alternative form of Rodrigues’ formula, 

P _ℓ  (m) =  1/ (2 ^l    ℓ!)  d ² / d m ²  ( m ²   -   1)  ^l 

 

   Then:  P ₀  =  1                                                                              

P ₁  =   m  

                                                                              

P ₂  =   3 m  ²  /2    -   ½

P ₃ =   5  m  ²  /2     -  3 m  /2

Meanwhile, the tesseral harmonics:

P m _ℓ  (m) =  (sin m f)

                      (cos  mf)  

Will vanish along the  2m meridians.  The parallels and meridians on which some sample harmonics, e.g.  P ³ ₃  (m )cos 3 q  vanish. As an example we can examine the zonal harmonic with ℓ  = 2,   m = 0 below.  

This would be done using the associated Legendre polynomial function given by:

P _ℓm (q )  =  (1 – z²) ^m/2 / ℓ! 2^ℓ   d ^(ℓ+m) / dz^(ℓ+m)  (z²   -1) ^ℓ

But ℓ  =2  and  m = 0 ,  so two parallels of latitude, i.e. ℓ  - m =  2  -  0 = 2

And:

P _ℓm (q )     =  (1 – z²) ⁰ / 2! 2²   d⁽²⁾  / dz⁽²⁾   (z²   -1) ²

=   1/ 8   [d² / dz²   (z²   -1) ²]

Y _ℓm (q,j) =  1  ^m [2 ℓ +1 (ℓ - m)!/ 4n ((ℓ +m)!] ^½ P ^ℓ_m (z ) exp (i m j),

Where z =  cos q and  w = 2 pn  where n = n _nℓm is the frequency of oscillation of mode n, ℓ, m

Let us also note here that: j = ( ℓ  - m)  and this brings us to the Laplace equation:

Ñ ²  F  =  

 ¶ ² / ¶ q ²   + (cos q / sin  q)  ¶  / ¶ q   +  1/ sin² q (¶ ² / ¶ f  ² )

And the possible eigenvalues  of  Ñ ² turn out to be the numbers:   

 – j (j + 1) for j = 0, 1, 2, 3……so that:

-          Ñ ²  F  =  - j (j + 1) F

Where  F   is the corresponding eigenfunction.  These eigenfunctions, e.g.

 F = 1, for j = m = 0

F = cos q (for j = 1, with m = 0)

F =  exp(+  if) sin q (for j = 1, with m = + 1))

F = 3 cos ²  q  - 1   (for j = 2, m = 0)

Are the spherical harmonics and it is standard to demand that these harmonics double  as the eigenfunctions of the operator

 ¶  / ¶ f  

Which commutes with  Ñ ²  and for which we have:

   ¶ F / ¶ f   =  imF

Where the possible eigenvalues are imaginaary (im) with the integer m in  the range:

 -j  <  m   <  j

From the preceding we see the eigenvalues, i.e. j(j + 1) are consonant with those for the total angular momentum in quantum mechanics,

J ²      =   L ₁ ²  +     L ₂ ²      +     L ₃ ²

Such that:  J ²     =   - Ñ ²   And: L ₃  = -i ¶  / ¶ f  

Any given combination of the numbers n, m and ℓ  allows a unique frequency n to be computed. For example, if we have n= 14, m = 16 and ℓ = 20 one gets a period of 340.61 s  which would be peculiar to solar frequencies.

Radial oscillations alone have ℓ  =  0 and we see in this case the associated Legendre function (P _ℓm (q )) has:

P _ℓm (q )  =  (1 – z²) ^m/2 / ℓ! 2^ℓ   d ^(ℓ+m) / dz^(ℓ+m)  (z²   -1) ^ℓ

Recall m= 2 ℓ + 1 = 2(0) +1 = 1

So:

P _ℓm (q )  =  (1 – z²) ^1/2 / 0! 2⁰   d / dz  (z²   -1) ⁰

=  (1 – z²) ^½  =   (1 – cos ² q)^½   =  (sin ² q)^½     = sin q

For q = p/2  , P _ℓm (q )  =  1

And: P _ℓm (q ) exp (i m j)  =  (1) exp (i (1) 0)  = 1

If  n _nℓm =   1 c/s  then:  Y _ℓm (q,j) = 1 and y _nℓm  =  R _n (r)  

Suggested Problems:

1) Suppose the solar p-modes create a surface pattern of nodes and waves such as shown below:

 

Use this diagram to deduce the values for ℓ  and m. 

2) Explain the appearance of the spherical surface shown below if it describes m= 5 and ℓ = 5. Thence, find the associated Legendre function: P _ℓm (q ) and also Y _ℓm (q,j) assuming q = p/2: (Assume  n _nℓm =  1 /s)