A computergenerated image of some of the 10 million modes of acoustic waves on the Sun (red indicating receding wave fronts, and blue approaching).
The phenomenon and measurement interpretation of solar oscillations being a relatively new
field, mainly developed in the last fortyfive years, few lay folk are aware of it.. However, plumbing the Sun's depths to investigate its different modes of vibration allows hitherto unknown
tools to be applied to many types of solar predictions.
The first reckoning of nonradial
oscillations arrived ca. 1968 with the work of Frazier who made twodimensional
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 "pmodes" 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 pmodes 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 pmodes in the
solar oscillations context, is 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:
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 oddnumbered 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, appear in the graphic
below:
As an example we can examine the zonal harmonic with ℓ = 2, m = 0, on the left.
This would be done using the associated Legendre polynomial function given by:
P _{ℓm} (q ) = (1 – z^{2})^{ m/2} / ℓ! 2^{ℓ} d^{ (ℓ+m)} / dz^{(ℓ+m)} (z^{2} 1)^{ ℓ}
But ℓ =2 and m =
0 , so:
P _{ℓm} (q ) = (1 – z^{2})^{ 0} / 2! 2^{2} d^{(2)}
^{ }/ dz^{(2)} (z^{2} 1)^{
2}
The completion of the analysis is left to the energetic reader and is given as a comprehension problem at the end.
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/ sin^{2} 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 eigenfunctons, 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 eigenfuynctions of the operator
¶ / ¶ f
Which commutes with Ñ ^{2
}and for which we have:
¶ F / ¶ f = imF
Where
the possible eigenvalues are (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 – z^{2})^{ m/2} / ℓ! 2^{ℓ} d^{ (ℓ+m)} / dz^{(ℓ+m)} (z^{2} 1)^{ ℓ}
Recall m= 2 ℓ + 1 = 2(0) +1 = 1
So:
P
_{ℓm}
(q ) = (1 –
z^{2})^{ 1/2} / 0! 2^{0} d^{
}/ dz (z^{2} 1)^{ 0}
= (1 – z^{2})^{ 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)
For practical assessment of solar vibrations we use k w diagrams with w along the ordinate and k (the wave number vector) along the abscissa. It is of interest to note that only waves with
the longest horizontal wavelength ℓ reach the core of the
Sun while high ℓmodes 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, one would see the pmodes in the upper left
lying above w_{ ac} and the gmodes (gravity modes) at lower
right below a dotted line for N. Another line given is for ck_{h} which
represents the Lamb waves or fmodes.
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
pmode 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 pmode
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}
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.
Ultimately the problem of the solar 5minute 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 c _{s }equals the horizontal phase velocity (w/k _{h} ) one
expects acoustic wave reflection.
Duvall and Harvey[2] reinforced this work by measuring the
frequency spectrum of this » 300s
oscillation and found it applicable for ℓmodes
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 +
4p^{2} g^{2} r^{2} / 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.
Comprehension Problems:
1) Take the ratio of specific heats g = 5/3 and the acoustic speed c _{s } = 900 m/s
with r
= 2650 km.
Find the ℓ =value using the Duvall –Harvey equation and explain why it does not make physical sense.
2)For the spherical surface with mode values m = 0 , ℓ = 2, find the
associated Legendre function: P _{ℓm} (q )
3) Explain
the basis for the spherical surface with the mode values m, ℓ shown below, and
find the associated Legendre function: (P _{ℓm} (q )):
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