Difference between revisions of "Spherical harmonic"

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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 &#x003bc; and longitude &#x003bb;:  <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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<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 &#x003bc; and longitude &#x003bb;:  <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:
ams2001glos-Se46
where Pm,n is the associated Legendre function defined as
ams2001glos-Se47
The spherical harmonic basis functions satisfy the orthogonal relationship
ams2001glos-Se48
and they satisfy the elliptic equation on the sphere:
ams2001glos-Se49