A Look At 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).

It is incomprehensible to many people that the Sun’s surface  could be undergoing vibrations in frequency that could be used as a diagnostic tool. This may be due to the phenomenon of solar oscillations being a relatively new field, mainly developed in the last forty years. It allows hitherto unknown tools to be applied to solar observations and predictions. In this chapter we will survey the basic physics associated with these observations.

The first reckoning of non-radial 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 here most applicable to the p-modes in the solar oscillations context, is given by:

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

And for normalized spherical harmonics:

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. (Note: j is measured from the meridian of the ascending node of the Sun’s equator.)

Thus, R _n (r)   is applicable to radial patterns (with n the radial quantum number) whereby for a given value of n we elicit a pattern of radial nodes, for which the position is determined by the exact pattern of the function  R _n (r)   The rest of y  gives the surface pattern as a spherical harmonic of the oscillation. 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. 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 or:

n =  2 p/ T  =  2 p/ (340.61 s) =  2.935 x 10 ^-3 /s

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) 

The degree ℓ of the spherical harmonic can assume any integer value, i.e.:

ℓ  =  0, 1, 2, …….

At each such ℓ the azimuthal number m assumes a 2 ℓ + 1 value, i.e.

m= - ℓ, (-ℓ +1)....0......( ℓ - 1), + ℓ

Meanwhile, the frequency of a particular mode is given by the azimuthal eigenvalue m, and the meriodonal eigenfunction ℓ - together with n. Since m= 2 ℓ + 1, then the spherical surface is split into 2 ℓ + 1  regions.

We are most interested in the Lamb frequency, e.g.

L   =   Ö( ℓ(ℓ + 1) c _s ² / r² )

And the Brunt- Vaisala or B-V frequency:

N ² =  g [G1 (d ln p _o  /dr) - d ln r _o  /dr) ]

Which can be simplified to:

N ² =  g² / c _s²  (g  - 1)

(Which holds expressly for the isothermal case.)

Where  c _s ²  = G1 p _o  /r _o    and:

G1= d (n lnp )  / ln r

Example Problem (1):

Find the Brunt- Vaisala frequency for the isothermal case if the sound speed in the Sun is c _s = 900 m/s. (Take g =  5/3 and g =  273 ms ^-2  )

Solution:

N ² =  g² / c _s ²  (g  - 1)

N =  Ö  (273 ms ^-2 /900 m/s) [ 5/3 – 1] ) = 0.372 /s

 Example Problem (2):

Find the Lamb frequency for the same sound speed, if  ℓ = 100 and r = 2650 km.

Solution:

We use:  L   =   Ö ( ℓ(ℓ + 1) c ² / r² )

Where r = 2650 km   = 2.65 x  10 ⁶  m  

L   =    Ö100(100 + 1) (900 m/s) ² / (2.65 x  10 ⁶  m ) ²

L   =  3.4  x  10 ^-2 / s   = 340 microhertz

    It is of interest to note that only waves with the longest horizontal wavelength ℓ  reach the core of the Sun while high ℓ-modes do not penetrate the convective zone. Given  ℓ = 100 we can expect the example chosen will be near the solar surface.  Also of interest in this context is the acoustic cutoff frequency, defined:

w _ac   = g g/ 2c

     This should not be confused with the plasma cutoff frequency, e.g. w _c     for EM waves in plasmas. But there is one common attribute for both: a cutoff frequency is for any frequency for which the wave number k ® 0.

In typical k -w diagrams, with frequency  w along the ordinate and k along the abscissa. One would see the p-modes in the upper left lying above  w _ac     and the g-modes (gravity modes) at lower right below a dotted line for N. Another line given is for ck_h which represents the Lamb waves or f-modes.    How many total modes, with n, ℓ and m distinct operate in the Sun? This is not difficult to estimate. Let’s take n first. According to diagnostic diagrams showing “ridges" for oscillatory power at each frequency at least 20 have been observed. In the diagram shown below, the spikes or ridges for the p-mode represent the first harmonic and the baseline smooth curve from which they project is the fundamental.

This leads to a maximum radial order of n = 20 for the p-mode associated ridges.. Now, for each of these n values, at least 500 angular degrees ℓ have been observed. We also know that for each such ℓ there are at least 2 ℓ values (actually 2 ℓ + 1). So in this case: 2 ℓ = 2(500) = 1000. Then the total estimated modes at any given time works out to:

T _{n ℓm}   =  20 x 500 x 1000 =  10 ⁷

 Or, ten million modes, all overlapping in time and space. Because of this extreme multiplicity of modes, photographs of the solar surface appear featureless or more accurately like a disk of coarse sandpaper. Bear in mind also what we’d observe at the solar surface are reflections of the multitude of standing sound waves (p-waves) that fill the Sun’s interior. Not surprisingly, these match up quite well with the coarse solar granulation, e.g

However, it has been Dopplergrams making use of the Michelson Doppler Imager (MDI) that have given us the best observation portal on the rising and falling super cells known as supergranules.

It was Robert Leighton – of Babcock and Leighton solar dynamo fame- who first observationally isolated the motions of solar granules and tied their delay time for replacement with an average lifetime of five minutes. He found a given granule’s velocity was equal and opposite to  the velocity at the same location some 2.5 minutes later. He deduced that something on the Sun must be oscillating. The convective cells don’t normally do so thus it had to be explained in an alternative way.

Ultimately the problem of the solar 5-minute oscillations was resolved by treating the Sun as a resonant cavity.  In 1981, Leibacher and Stein showed that if one treated the Sun as a resonant cavity one could expect the relationship:

T = (n + ½)p / w

     In other words for the condition at which the sound speed equals the horizontal phase velocity (w/k _h ) one expects acoustic wave reflection.

Duvall and Harvey[1]  reinforced this work by measuring the frequency spectrum of this » 300s  oscillation and found it applicable for ℓ-modes less than 140, and radial modes R with order n = 2 to 26. Posing the degree ℓ- in terms of  the reflection radius r:

ℓ =  -1/2  +  [ ¼ +    4p² g² r² / c _s ² ]

The modes were thus established as being deep in the solar interior by matching all the modes in a series of data using the above equation.

Problems:

1) Find the acoustic cutoff frequency below which all waves would be reflected if a sound speed of 900 m/s applies.

2) Sketch the spherical surface with the mode values m = 0 and  ℓ = 2, and justify your sketch. Find the associated Legendre function:        (P _ℓm (q )).

3) Let a period T= 340.61 s be associated with the Lamb frequency. What would be the associated depth of reflection in the Sun, assuming ℓ = 20 and c= 900 m/s.

4) For the p-mode pattern shown find the associated Legendre function:  (P _ℓm (q )) if If q = p/4.

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[1] Duvall, T.L. and Harvey, J.W.: 1984, Nature, 310, 19.