# Difference between revisions of "Power-law profile"

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− | <div class="definition"><div class="short_definition">A formula for the [[variation]] of [[wind]] with height in the [[surface boundary layer]].</div><br/> <div class="paragraph">It is an alternative to the [[logarithmic velocity profile]], and the assumptions are the same, with the exception of the form of the dependence of [[mixing length]] ''l'' on height ''z''. Here <div class="display-formula"><blockquote>[[File:ams2001glos-Pe44.gif|link=|center|ams2001glos-Pe44]]</blockquote></div> Then <div class="display-formula"><blockquote>[[File:ams2001glos-Pe45.gif|link=|center|ams2001glos-Pe45]]</blockquote></div> where <div class="inline-formula">[[File:ams2001glos-Pex07.gif|link=|ams2001glos-Pex07]]</div> is the [[mean velocity]], ''u''<sub>*</sub> the [[friction velocity]], ν the [[kinematic viscosity]], and <div class="display-formula"><blockquote>[[File:ams2001glos-Pe46.gif|link=|center|ams2001glos-Pe46]]</blockquote></div> For moderate [[Reynolds numbers]], ''p'' = 6/7 (the seventh-root profile) is empirically verified, but for large Reynolds numbers ''p'' is between this value and unity. It is to be noted that if <div class="inline-formula">[[File:ams2001glos-Pex08.gif|link=|ams2001glos-Pex08]]</div> is proportional to ''z''<sup>''m''</sup>, and if the [[stress]] is assumed independent of height, then the [[eddy viscosity]] ν<sub>''e''</sub> is proportional to ''z''<sup>1 | + | <div class="definition"><div class="short_definition">A formula for the [[variation]] of [[wind]] with height in the [[surface boundary layer]].</div><br/> <div class="paragraph">It is an alternative to the [[logarithmic velocity profile]], and the assumptions are the same, with the exception of the form of the dependence of [[mixing length]] ''l'' on height ''z''. Here <div class="display-formula"><blockquote>[[File:ams2001glos-Pe44.gif|link=|center|ams2001glos-Pe44]]</blockquote></div> Then <div class="display-formula"><blockquote>[[File:ams2001glos-Pe45.gif|link=|center|ams2001glos-Pe45]]</blockquote></div> where <div class="inline-formula">[[File:ams2001glos-Pex07.gif|link=|ams2001glos-Pex07]]</div> is the [[mean velocity]], ''u''<sub>*</sub> the [[friction velocity]], ν the [[kinematic viscosity|kinematic viscosity]], and <div class="display-formula"><blockquote>[[File:ams2001glos-Pe46.gif|link=|center|ams2001glos-Pe46]]</blockquote></div> For moderate [[Reynolds numbers]], ''p'' = 6/7 (the seventh-root profile) is empirically verified, but for large Reynolds numbers ''p'' is between this value and unity. It is to be noted that if <div class="inline-formula">[[File:ams2001glos-Pex08.gif|link=|ams2001glos-Pex08]]</div> is proportional to ''z''<sup>''m''</sup>, and if the [[stress]] is assumed independent of height, then the [[eddy viscosity]] ν<sub>''e''</sub> is proportional to ''z''<sup>1-''m''</sup>. These relations are known as Schmidt's conjugate-power laws.</div><br/> </div><div class="reference">Sutton, O. G. 1953. Micrometeorology. 78–85. </div><br/> |

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## Latest revision as of 17:39, 25 April 2012

## power-law profile

It is an alternative to the logarithmic velocity profile, and the assumptions are the same, with the exception of the form of the dependence of mixing length Then where is the mean velocity, For moderate Reynolds numbers,

*l*on height*z*. Here*u*_{*}the friction velocity, ν the kinematic viscosity, and*p*= 6/7 (the seventh-root profile) is empirically verified, but for large Reynolds numbers*p*is between this value and unity. It is to be noted that if is proportional to*z*^{m}, and if the stress is assumed independent of height, then the eddy viscosity ν_{e}is proportional to*z*^{1-m}. These relations are known as Schmidt's conjugate-power laws.Sutton, O. G. 1953. Micrometeorology. 78–85.