Spherical harmonic: Difference between revisions
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An analytic basis function on the sphere that is commonly used in a [[spectral model|spectral model]].<br/> A spherical harmonic is defined for each total [[wavenumber]] ''n'' and [[zonal wavenumber]] ''m'' as the following function of sine of latitude μ and longitude λ: <blockquote>[[File:ams2001glos-Se46.gif|link=|center|ams2001glos-Se46]]</blockquote> where ''P''<sub>''m'',''n''</sub> is the associated Legendre function defined as <blockquote>[[File:ams2001glos-Se47.gif|link=|center|ams2001glos-Se47]]</blockquote> The spherical harmonic basis functions satisfy the [[orthogonal]] relationship <blockquote>[[File:ams2001glos-Se48.gif|link=|center|ams2001glos-Se48]]</blockquote> and they satisfy the elliptic equation on the sphere: <blockquote>[[File:ams2001glos-Se49.gif|link=|center|ams2001glos-Se49]]</blockquote> | |||
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Latest revision as of 07:46, 30 March 2024
An analytic basis function on the sphere that is commonly used in a spectral model.
A spherical harmonic is defined for each total wavenumber n and zonal wavenumber m as the following function of sine of latitude μ and longitude λ:
A spherical harmonic is defined for each total wavenumber n and zonal wavenumber m as the following function of sine of latitude μ and longitude λ:
where Pm,n is the associated Legendre function defined as
The spherical harmonic basis functions satisfy the orthogonal relationship
and they satisfy the elliptic equation on the sphere:
