Revisiting the formulations for the longitudinal velocity variance in the unstable atmospheric surface layer. (27th November 2014)

• Record Type: Journal Article
• Authors: Banerjee, T.; Katul, G. G.; Salesky, S. T.; Chamecki, M.
• Is Part Of: Quarterly journal of the Royal Meteorological Society. Volume 141:Number 690(2015:Jul.)
• Page Start: 1699
• Page End: 1711
• Publication Date: 2014-11-27
• Subjects: Meteorology -- Periodicals; 551.5
• DOI: 10.1002/qj.2472
• Languages: English
• ISSNs: 0035-9009

Abstract

<abstract abstract-type="main" id="qj2472-abs-0001"> <title> <x xml:space="preserve">Abstract</x> </title> <p id="qj2472-para-0001">Because of its non‐conformity to Monin–Obukhov Similarity Theory (MOST), the effects of thermal stratification on scaling laws describing the streamwise turbulent intensity <italic>σ</italic><sub><italic>u</italic></sub> normalized by the turbulent friction velocity (<italic>u</italic><sub>∗</sub>) continue to draw research attention. A spectral budget method has been developed to assess the variability of <italic>σ</italic><sub><italic>u</italic></sub>/<italic>u</italic><sub>∗</sub> under unstable atmospheric stratification. At least three different length‐scales—the distance from the ground (<italic>z</italic>), the height of the atmospheric boundary layer (<italic>δ</italic>) and the Obukhov length (<italic>L</italic>)–are all found to be controlling parameters in the variation of <italic>σ</italic><sub><italic>u</italic></sub>/<italic>u</italic><sub>∗</sub>. Analytical models have been developed and supported by experiments for two limiting conditions: <italic>z</italic>/<italic>δ</italic> < 0.02, −<italic>z</italic>/<italic>L</italic> < 0.5 and 0.02 ≪ <italic>z</italic>/<italic>δ</italic> < 0.1, −<italic>z</italic>/<italic>L</italic> > 0.5. Under the first constraint, the turbulent kinetic energy spectrum is predicted to follow three regimes: <italic>k</italic><sup>0</sup>, <italic>k</italic><sup>−1</sup> and <italic>k</italic><sup>−5/3</sup>, divided in the last two regimes by a break‐point at <italic>kz</italic> = 1, where <italic>k</italic> denotes the wave number. The quantity <italic>σ</italic><sub><italic>u</italic></sub>/<italic>u</italic><sub>∗</sub> is shown to follow the much discussed logarithmic scaling, reconciled to Townsend's attached eddy hypothesis <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj2fnsc1fd" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:qj:media:qj2472:qj2472-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>/</mml:mo><mml:msubsup><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mo>∗</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mtext>log</mml:mtext><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>/</mml:mo><mml:mi>δ</mml:mi><mml:mo>)</mml:mo></mml:math></alternatives></inline-formula>, where the coefficients <italic>B</italic><sub>1</sub> and <italic>A</italic><sub>1</sub> are modified by MOST for mildly unstable stratification. Under the second constraint, the turbulent energy spectrum tends to become quasi‐inertial, displaying <italic>k</italic><sup>0</sup> and <italic>k</italic><sup>−5/3</sup> with a break‐point predicted to occur at 0.3 < <italic>kz</italic> < 1. The work here brings together well‐established but seemingly unrelated theories of turbulence such as Kolmogorov's hypothesis, Townsend's attached eddy hypothesis, MOST and Heisenberg's eddy viscosity under a common framework.</p> </abstract>