Revisiting the formulations for the longitudinal velocity variance in the unstable atmospheric surface layer

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 σu2/u*2=B1-A1log(z/δ), 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.

Duke Scholars

Author Gabriel G. Katul Civil and Environmental Engineering