Plots of the Spherical Harmonics Eigenmodes

Written 2020 Jul 19 — Tag: Spherical Harmonics

This posting is a simple addendum to the much longer article, Notes on Calculating the Spherical Harmonics. Because it had already grown so long, I left out making any plots of individual eigenmodes, so here I simply present plots of the first 50 degrees of the pure spherical harmonic eigenmodes, which can be useful to help visualize some of the symmetry properties that were discussed in the longer article.

Introduction

As a reminder, the spherical harmonics eigenmodes (or eigenfunctions) are the set of functions

\begin{align} Y_{\ell m} (\theta, \phi) &\equiv \lambda_\ell^m(\cos \theta) e^{im\phi} \end{align}

where the functions are indexed by two integers — the degree $\ell$ and order $m$ — and are functions of the colatitude-azimuth angular position $(\theta,\phi)$ on the surface of the sphere. The functions $\lambda_\ell^m(x)$ are the spherical-harmonic pre-normalized Associated Legendre Polynomials, which are the subject of my Legendre.jl article series.

The following figure is an interactive “figure pager”1 which shows the real part of a spherical harmonic eigenmode (for an orthographic projection of the sphere) and the corresponding degree $\ell$ and order $m$ labeled at the top of the image and in the textbox to the left of the image. You can either use the arrow buttons to step through the eigenmodes, or you can type in a degree and order into the textbox — formatted like “(50,25)” where the first number is the degree and the second is the order — to jump to any specific eigenmode within the range $0 \le \ell \le 50$ and $0 \le m \le \ell$.

Orthographic projection of the spherical harmonics
Orthographic projection of the real part of the spherical harmonics (where the camera view is centered on 30°N, 150°W). The color scale is peak-normalized, so the relative mode-to-mode amplitudes are not comparable. The imaginary part is similar and differs by a relative azimuthal rotation.

For instance, these plots help make the naming of the zonal, tesseral, and sectoral spherical harmonics a bit clearer: the zonal harmonics are those where $m = 0$ and the eigenmode is azimuthally symmetric around the poles; the sectoral harmonics are those where $m = \ell$ and the plot looks a bit like the sectors of an orange (no latitude lines of zero); and the tesseral harmonics for all other values of $m$ where the pattern looks more like a checkerboard. (I suggest you try out some of these combinations in the pager above.)

I also created a movie which runs through all 1326 eigenmodes in increasing order and degree.

Similar to Figure 1.1, but the entire series of images have been rendered into a movie which steps through all eigenmodes, from low to high order and degree.

  1. The figure pager is built using a small, self-contained Javascript library that I previously wrote for use in my graduate research when writing up explanations of whatever work I was doing. The library is freely available on my Github for use by anyone. ↩︎