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>&#x0002a;</sub> the [[friction velocity]], &#x003bd; 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]]  &#x003bd;<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&ndash;85. </div><br/>  
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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>&#x0002a;</sub> the [[friction velocity]], &#x003bd; 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]]  &#x003bd;<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&ndash;85. </div><br/>  
 
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Latest revision as of 17:39, 25 April 2012



power-law profile

A formula for the variation of wind with height in the surface boundary layer.

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
ams2001glos-Pe44
Then
ams2001glos-Pe45
where
ams2001glos-Pex07
is the mean velocity, u* the friction velocity, ν the kinematic viscosity, and
ams2001glos-Pe46
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
ams2001glos-Pex08
is proportional to zm, and if the stress is assumed independent of height, then the eddy viscosity νe is proportional to z1-m. These relations are known as Schmidt's conjugate-power laws.

Sutton, O. G. 1953. Micrometeorology. 78–85.