Spherical Harmonics: Degree l & Order m Structure & Variation

[henrybrent]

I am studying the Earths main magnetic field (internal, specifically the stuff at the Core-Mantle boundary) which has led me to spherical harmonics. I am curious... how is the structure of a spherical harmonic determined by its degree l and order m? What role do the first three coefficients have (are they known as Gaussian Coefficients?) , and the variation in structure within a particular degree l.

I have these questions as I have been playing around with a model of the magnetic field, changing these coefficients but I am at a loss and see no pattern occurring, however, this question applies just to spherical harmonics in general so I can gauge a more overall understanding.

 

[blue_leaf77]

Spherical harmonics usually arise when there is a spherical symmetry in the system, or when the system will become easier to describe if spherical coordinate were used.

how is the structure of a spherical harmonic determined by its degree l and order m?

You have to look at the functional form of the spherical harmonic corresponding to the particular values of l and m. l characterizes the system's functional dependence on polar angles (I guess in the sense of Earth's coordinate it would be related to the latitude), while m is related to the dependence on azimuthal angle (longitude).

first three coefficients

Three? what are they?

 

[henrybrent]

Spherical harmonics usually arise when there is a spherical symmetry in the system, or when the system will become easier to describe if spherical coordinate were used.

You have to look at the functional form of the spherical harmonic corresponding to the particular values of l and m. l characterizes the system's functional dependence on polar angles (I guess in the sense of Earth's coordinate it would be related to the latitude), while m is related to the dependence on azimuthal angle (longitude).

Three? what are they?

I am not sure how to notate them properly, but g1, g2, g3?

 

[Khashishi]

Spherical harmonics are a 2D extension to Fourier harmonics. Each term in the Fourier series is called a harmonic. One way of thinking about a periodic function is to see it as a function defined on a circle. The circle loops back on itself, hence the periodicity. A function defined on a sphere is periodic in 2D, so you need something more complex, so the harmonics are parametrized by two values, l and m, instead of just one for Fourier.

There are different choices of basis you can use for Fourier harmonics. You can use sines and/or cosines, or you can use exponentials with imaginary arguments. The form with exponentials ties in directly to the spherical harmonics. In fact, if you think of a 1D periodic function as being defined on the equator of a sphere, then the nth Fourier harmonic (in exponential form) is completely analogous to the spherical harmonic with m = n. The m value of the spherical harmonic tells you the frequency of the oscillation going around the equator (equivalent to the Fourier harmonic). m can be positive or negative. $l$ is the magnitude of the frequency of the oscillation in all directions. It's hard to visualize $l$ but if you define $k = l - |m|$, then $k$ is the frequency of oscillation in the north/south direction. The north/south direction is treated differently than the east/west direction due to our choice of coordinates. You can't go north of the north pole, but you can keep going east as far as you like.

 

[Khashishi]

There's another thread right here https://www.physicsforums.com/threads/spherical-harmonics-color-map.805216/

 

[henrybrent]

Spherical harmonics are a 2D extension to Fourier harmonics. Each term in the Fourier series is called a harmonic. One way of thinking about a periodic function is to see it as a function defined on a circle. The circle loops back on itself, hence the periodicity. A function defined on a sphere is periodic in 2D, so you need something more complex, so the harmonics are parametrized by two values, l and m, instead of just one for Fourier.

There are different choices of basis you can use for Fourier harmonics. You can use sines and/or cosines, or you can use exponentials with imaginary arguments. The form with exponentials ties in directly to the spherical harmonics. In fact, if you think of a 1D periodic function as being defined on the equator of a sphere, then the nth Fourier harmonic (in exponential form) is completely analogous to the spherical harmonic with m = n. The m value of the spherical harmonic tells you the frequency of the oscillation going around the equator (equivalent to the Fourier harmonic). m can be positive or negative. $l$ is the magnitude of the frequency of the oscillation in all directions. It's hard to visualize $l$ but if you define $k = l - |m|$, then $k$ is the frequency of oscillation in the north/south direction. The north/south direction is treated differently than the east/west direction due to our choice of coordinates. You can't go north of the north pole, but you can keep going east as far as you like.

I still don't quite get how changing L and M would affect the structure of the harmonic ?

I have manipulated L and M, and in the case of the Earth,adds more lines, a bit like latitudes, and then the changing M distorts them? any idea of the gaussian co-efficients?

 

[Khashishi]

Where are you seeing Gaussian coefficients? Maybe if you cite what you are looking at...
It sounds like you are using a program to display l, m. What program is that?

 

[henrybrent]

Where are you seeing Gaussian coefficients? Maybe if you cite what you are looking at...
It sounds like you are using a program to display l, m. What program is that?

It's some UNIX environment, Hummingbird Exceed/Open Exceed?

 

[Khashishi]

I've never seen "Gauss coefficients" before, but it's pretty clear what they are talking about from the image. I said before that Fourier series can have either sine/cosine basis or exponential basis. The same is true for spherical harmonics, although I have never actually seen sin/cosine used for the basis before this example. In physics, imaginary exponentials are usually used.

The g and h coefficients are the amplitudes of each spherical harmonic. Just like how in a Fourier series, your periodic function is built up out of some amplitude of each sine/cosine harmonic.

Do you understand Fourier series? Here I'm trying to explain how spherical harmonics are just 2D Fourier series, but it doesn't help if you don't understand Fourier series.

 

[henrybrent]

I've never seen "Gauss coefficients" before, but it's pretty clear what they are talking about from the image. I said before that Fourier series can have either sine/cosine basis or exponential basis. The same is true for spherical harmonics, although I have never actually seen sin/cosine used for the basis before this example. In physics, imaginary exponentials are usually used.

The g and h coefficients are the amplitudes of each spherical harmonic. Just like how in a Fourier series, your periodic function is built up out of some amplitude of each sine/cosine harmonic.

Do you understand Fourier series? Here I'm trying to explain how spherical harmonics are just 2D Fourier series, but it doesn't help if you don't understand Fourier series.

Yes, yes. I'm going about asking my question in the incorrect way I suppose..

I am trying to work out how does changing M and L affect the structure, like in the image

 

[henrybrent]

here is a better image to describe what I am trying to work out

 

[Khashishi]

l-m is the number of nodes in the vertical direction.
2m is the number of nodes in the horizontal direction. That's because these nodes have to come in pairs since they are great circles.