Revisiting the formulations for the longitudinal velocity variance in the unstable atmospheric surface layer. (27th November 2014) - Banerjee, T.; Katul, G. G.; Salesky, S. T.; Chamecki, M.
<x xml:space="preserve">Abstract</x> Because of its non‐conformity to Monin–Obukhov Similarity Theory (MOST), the effects of thermal stratification on scaling laws describing the streamwise turbulent intensity σu normalized by the turbulent friction velocity (u∗) continue to draw research attention. A spectral budget method has been developed to assess the variability of σu/u∗ under unstable atmospheric stratification. At least three different length‐scales—the distance from the ground (z), the height of the atmospheric boundary layer (δ) and the Obukhov length (L)–are all found to be controlling parameters in the variation of σu/u∗. Analytical models have been developed and supported by experiments for two limiting conditions: z/δ < 0.02, −z/L < 0.5 and 0.02 ≪ z/δ < 0.1, −z/L > 0.5. Under the first constraint, the turbulent kinetic energy spectrum is predicted to follow three regimes: k0, k−1 and k−5/3, divided in the last two regimes by a break‐point at kz = 1, where k denotes the wave number. The quantity σu/u∗ is shown to follow the much discussed logarithmic scaling, reconciled to Townsend's attached eddy hypothesis <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj2fnsc1fd" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />σu2/u∗2=B1−A1log(z/δ)</alternatives>, where the coefficients B1 and A1 are modified by MOST for mildly unstable stratification. Under the second constraint, the turbulent energy spectrum tends to become quasi‐inertial, displaying k0 and k−5/3 with a break‐point predicted to occur at 0.3 < kz < 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.