Spherical harmonic: Difference between revisions
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<div class="definition"><div class="short_definition">An analytic basis function on the sphere that is commonly used in a [[spectral model]].</div><br/> <div class="paragraph">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: <div class="display-formula"><blockquote>[[File:ams2001glos-Se46.gif|link=|center|ams2001glos-Se46]]</blockquote></div> where ''P''<sub>''m'',''n''</sub> is the associated Legendre function defined as <div class="display-formula"><blockquote>[[File:ams2001glos-Se47.gif|link=|center|ams2001glos-Se47]]</blockquote></div> The spherical harmonic basis functions satisfy the [[orthogonal]] relationship <div class="display-formula"><blockquote>[[File:ams2001glos-Se48.gif|link=|center|ams2001glos-Se48]]</blockquote></div> and they satisfy the elliptic equation on the sphere: <div class="display-formula"><blockquote>[[File:ams2001glos-Se49.gif|link=|center|ams2001glos-Se49]]</blockquote></div></div><br/></div> | <div class="definition"><div class="short_definition">An analytic basis function on the sphere that is commonly used in a [[spectral model|spectral model]].</div><br/> <div class="paragraph">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: <div class="display-formula"><blockquote>[[File:ams2001glos-Se46.gif|link=|center|ams2001glos-Se46]]</blockquote></div> where ''P''<sub>''m'',''n''</sub> is the associated Legendre function defined as <div class="display-formula"><blockquote>[[File:ams2001glos-Se47.gif|link=|center|ams2001glos-Se47]]</blockquote></div> The spherical harmonic basis functions satisfy the [[orthogonal]] relationship <div class="display-formula"><blockquote>[[File:ams2001glos-Se48.gif|link=|center|ams2001glos-Se48]]</blockquote></div> and they satisfy the elliptic equation on the sphere: <div class="display-formula"><blockquote>[[File:ams2001glos-Se49.gif|link=|center|ams2001glos-Se49]]</blockquote></div></div><br/></div> | ||
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Latest revision as of 17:57, 25 April 2012
spherical harmonic
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: 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:
