View source for Logarithmic velocity profile

== logarithmic velocity profile ==

The [[variation]] of the mean wind speed with height in the [[surface boundary layer]] derived with the following assumptions: 1) the mean motion is one-dimensional;   2) the [[Coriolis force]] can be neglected; 3) the [[shearing stress]] and [[pressure gradient]] are independent  of height; 4) the [[pressure force]] can be neglected with respect to the [[viscous force]]; and  5) the [[mixing length]] ''l'' depends only on the fluid and the distance from the boundary, ''l'' = ''kz''.<br/> Near aerodynamically smooth surfaces, the result is  <blockquote>[[File:ams2001glos-Le29.gif|link=|center|ams2001glos-Le29]]</blockquote> that is, the logarithmic velocity profile, where ''u''<sub>&#x0002a;</sub> is the [[friction velocity]] and &#x003bd; the [[kinematic  viscosity]]. ''k'' &cong; 0.4 and has been called the K&#x000e1;rm&#x000e1;n constant or [[Von_kármán%27s_constant|von K&#x000e1;rm&#x000e1;n's constant]]. The  equation fails for a height ''z'' sufficiently close to the surface. For aerodynamically rough flow,  [[molecular viscosity]] becomes negligible. The profile is then  <blockquote>[[File:ams2001glos-Le30.gif|link=|center|ams2001glos-Le30]]</blockquote>''z''<sub>0</sub> is a constant related to the average height &#x003b5; of the surface irregularities by ''z''<sub>0</sub> = &#x003b5;/30 and is  called the [[aerodynamic roughness length]]. Another derivation of the logarithmic profile was  obtained by Rossby under the assumption that for fully rough flow the roughness affects the [[mixing length|mixing  length]] only in the region where ''z'' and ''z''<sub>0</sub> are comparable. Then ''l'' = ''k''(''z'' + ''z''<sub>0</sub>) and  <blockquote>[[File:ams2001glos-Le31.gif|link=|center|ams2001glos-Le31]]</blockquote> For statically nonneutral conditions, a [[stability]] correction factor can be included (<br/>''see'' equation in  definition of [[aerodynamic roughness length]]).<br/> Haugen, D. A. 1973. Workshop on Micrometeorology. Amer. Meteor. Soc., 392 pp. <br/> Sutton, O. G. 1953. Micrometeorology. sect. 3.9. <br/>