Visualization of Spherical Harmonics

This is an interactive web page (based on Javascript) to visualize Spherical Harmonics. You can directly select the indices 'l' and 'm' or start an animation ('show all m' loops 'm' for fixed 'l' by 'm = 0,...,l'   and 'show 20 l' loops 'l' for fixed 'm' by 'l = m,...,m+20')
All other options 'back', 'rotate', 'export', 'extras',... are explained below the image.

For the use of spherical harmonics in global gravitational potential modelling, see the Gravity Field Tutorial (PDF). 1

Short Explanation
The image shows one specific surface spherical harmonic of degree 'l' and order 'm', denoted as 'Ylm(φ,λ)' (they are solutions of ΔYlm = 0).
Any source-free field on the sphere can be computed with these terms as the double sum over 'l' (l = 1,..,lmax) and 'm' (m = 0,..,l) with:

G(φ,λ) = ∑lm Ylm(φ,λ)   with   Ylm(φ,λ) = Plm(sin(φ)) × [Clm cos(m λ) + Slm sin(m λ)]

where 'φ' = [-90°,+90°] is the latitude, 'λ' = [0°,360°] the longitude. 'Plm(x)' are the Legendre polynoms and 'Clm', 'Slm' are constant coefficients, which describe the actual field/model.
Here the cos-term 'ClmPlm(sin(φ))cos(m λ)' is displayed, arbitrarily scaled by Clm = 100⁄sqrt(l). The sin-term is identical with a phase shift of π⁄m.
The blue lines indicate the zero-circles, where 'Ylm = 0'.

Select 'l' Select 'm'
Link to the old Java version

1 extracted from: Ilk, K.H., J. Flury, R. Rummel, P. Schwintzer, W. Bosch, C. Haas, J. Schröter, D. Stammer, W. Zahel, H. Miller, R. Dietrich, P. Huybrechts, H. Schmeling, D. Wolf, J. Riegger, A. Bardossy, A. Güntner, 2004: Mass Transport and Mass Distribution in the Earth System (PDF) -- Contribution of the New Generation of Satellite Gravity and Altimetry Missions to Geosciences, Proposal for a German Priority Research Program

    Wolfgang Köhler