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 sigma(u) normalized by the turbulent friction velocity (u) continue to draw research attention. A spectral budget method has been developed to assess the variability of sigma(u)/u under unstable atmospheric stratification. At least three different length-scalesthe distance from the ground (z), the height of the atmospheric boundary layer (delta) and the Obukhov length (L)-are all found to be controlling parameters in the variation of sigma(u)/u. Analytical models have been developed and supported by experiments for two limiting conditions: z/ < 0.02, -z/L < 0.5 and 0.02 << z/delta < 0.1, -z/L > 0.5. Under the first constraint, the turbulent kinetic energy spectrum is predicted to follow three regimes: k(0), 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 sigma(u)/u is shown to follow the much discussed logarithmic scaling, reconciled to Townsend's attached eddy hypothesis sigma(2)(u)/u(*)(2) = B-1 - A(1) log(z/delta), where the coefficients B-1 and A(1) are modified by MOST for mildly unstable stratification. Under the second constraint, the turbulent energy spectrum tends to become quasi-inertial, displaying k(0) 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.