Some questions about the PFSS model on IMF

[Tianluo_Qi]

Hey guys, I am a Chinese student who is now reading the doctoral dissertation written by Mr. Heoksema, Structure and Evolution of the Large-Scale Solar and Heliospheric Magnetic Fields, for the preparation of my bachelor thesis. But some puzzles arise while reading the mathematic development of the model in chapter three. If you happen to have read this thesis before, it will be very appreciated to shed some light on those puzzles for me. Thank you very much!

 

[Tianluo_Qi]

1.While changing the order of the summation of index $l$ and $m$ when dealing with the line-of-sight component of the photospheric field, the writer omit the minus part of m.(P45)

as we know, the index m range from $-l$ through zero to $l$ for one certian $l$ in the spherical harmonics function, so for the nonnegative part of m, the following equation is right:

$\sum^{\infty}_{l=0}\sum^{l}_{m=0}A_{lm}=\sum_{m}\sum^{\infty}_{l=m}A_{lm}$​

but, for the negative part, the right equation is:

$\sum^{\infty}_{l=1}\sum^{-1}_{m=-l}A_{lm}=\sum_{m}\sum^{\infty}_{l=-m}A_{lm}$​

in the dissertation of Mr. Hoeksema, the first equation is used, for which the only logical explanation I come up with is that he omit the negative part of m, but he doesn't explain why this ignorance is adequate.

 

[Tianluo_Qi]

2.The orthogonal coefficient the writer shows for the spherical harmonics function is not consistent with what I learn. (P46)
Of course the orthogonal coefficient will not affect the result for different $l,l^{'}$ or different $m,m^{'}$, so let’s focus on the equal ones. And as far as I know, the orthogonal coefficient of associated Legendre Polynomial is:

$\int_{0}^{\pi}sin\theta P_{l}^{m}P_{l}^{m}d\theta =\dfrac{(l+m)!}{(l-m)!}\dfrac{2}{2l+1}$​

And the integral of the phi part is (result being the same for sine and cosine):

$\int_{0}^{2\pi}cos^{2}m\phi d\phi=\pi$​

So I think the orthogonal coefficient for the spherical harmonics function should be:

$\dfrac{(l+m)!}{(l-m)!}\dfrac{2\pi}{2l+1}\delta_{ll^{'}}\delta_{mm^{'}}$​

but not what the writer shows:

$\dfrac{4\pi}{2l+1}\delta_{ll^{'}}\delta_{mm^{'}}$​

 

[Tianluo_Qi]

3.The coefficient before the sum-like integral expression is not correct even we assume that the orthogonal coefficient in the former page is correct. (P47)
If we refer to the orthogonal coefficient without Kronecker delta part as $OC_{lm}$, then the coefficient before the sum-like integral expression here should be:

$\dfrac{1}{OC_{lm}}\dfrac{2\pi}{N}\dfrac{\pi}{M}$​

So if the writer's result is correct, the coefficient here should be:

$\dfrac{2l+1}{NM}\dfrac{\pi}{2}$​

And on the contrary:

$\dfrac{(l-m)!}{(l+m)!}\dfrac{2l+1}{NM}\pi$​

Furthermore, I think the expression here within the summation notation should be $sin^{2}\theta_{i}$, but I am not sure if this is just a typo.

 

[Tianluo_Qi]

4.About the exact meaning of truncation for the calculation of $g_{l}^{m}$ and $h_{l}^{m}$.(P47)
I see the truncation at the maximum index $T$ here as we ordering $g_{l}^{m}=0(l>T)$(all the same for $h_{l}^{m}$, so I just discuss $g_{l}^{m}$ below), so all the equations for the same $m$ here are:

$\beta_{m}^{m}g_{m-2}^{m}+\alpha_{m}^{m}g_{m}^{m}+\gamma_{m}^{m}g_{m+2}^{m}=\alpha_{m}^{m}g_{m}^{m}+\gamma_{m}^{m}g_{m+2}^{m}=a_{mm}$
$\beta_{m+1}^{m}g_{m-1}^{m}+\alpha_{m+1}^{m}g_{m+1}^{m}+\gamma_{m+1}^{m}g_{m+3}^{m}=\alpha_{m+1}^{m}g_{m+1}^{m}+\gamma_{m+1}^{m}g_{m+3}^{m}=a_{(m+1)m}$
$\vdots$
$\beta_{T}^{m}g_{T-2}^{m}+\alpha_{T}^{m}g_{T}^{m}+\gamma_{T}^{m}g_{T+2}^{m}=\beta_{T}^{m}g_{T-2}^{m}+\alpha_{T}^{m}g_{T}^{m}=a_{Tm}$
$\beta_{T+1}^{m}g_{T-1}^{m}+\alpha_{T+1}^{m}g_{T+1}^{m}+\gamma_{T+1}^{m}g_{T+3}^{m}=\beta_{T+1}^{m}g_{T-1}^{m}=a_{(T+1)m}$
$\beta_{T+2}^{m}g_{T}^{m}+\alpha_{T+2}^{m}g_{T+2}^{m}+\gamma_{T+2}^{m}g_{T+4}^{m}=\beta_{T+2}^{m}g_{T}^{m}=a_{(T+2)m}$
$\vdots$​

The right side of all the expressions represented by the second ellipses is zero so we need not to care about them. But there are still two more equations which are not contained in the matrix the write shows, so my question is that why we should omit them. My partner says that for $T-m+1$ variables, $T-m+3$ equations may lead to contradictory of equations, so we should ignore the last two equations. But I think for the theoretical component $g_{l}^{m}$, of course we can order some of them to be zero, but as a number calculated out by one definite expression, we have no right to order $\beta_{l}^{m}$ to be zero. In my opinion, the only thing we can do to deal with this problem is to prove that the two more equations will not lead to contradictory of equations, but I have no clue how to do it.
And the expression for $\beta_{l}^{m}$ is:

$\beta_{l}^{m}=\dfrac{Q_{(l-1)}^{m}Q_{l}^{m}}{2l-1}$​

where $Q_{l}^{m}$ represents:

$Q_{l}^{m}=\sqrt{l^2-m^2}$​

 

[stefan r]

I tried opening the article link. It requires an account.

 

[stefan r]

https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19850002589.pdf . This looks like may be the same. Not sure if I will read it soon.

 

[Tianluo_Qi]

https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19850002589.pdf . This looks like may be the same. Not sure if I will read it soon.

yey, it is and the page number is the same too