Longitudinal Turbulent Velocity Variance Under Unstable and Stable Atmospheric Stratification Explained by a Spectral Budget

Because of its non-conformity to Monin-Obukhov Similarity Theory (MOST), the effects of thermal stratification on scaling laws describing the stream-wise turbulent intensity s_u normalized by the turbulent friction velocity u_* continues to draw research attention. The streamwise turbulent intensity is useful for a copious number of practical applications- ranging from industrial pipe flow to air pollution modeling to footprint determination in designing and interpreting measurements.  Hence, an analytical model able to predict its nature would benefit all the aforementioned applications. A spectral budget method used previously by Banerjee and Katul (2013) was demonstrated as a suitable workhorse to analytically explain the `universal' logarithmic scaling law exhibited by s_u²/u_*² for neutral conditions as reported in different high Reynolds number experiments. In the present work, that theoretical framework has been expanded to assess the variability of s_u/u_* under unstable atmospheric stratification (Banerjee et al., 2014). At least three different length scales- the distance from the ground (z), the height of the atmospheric boundary layer d, and the Obukhov length (L) are all found to be controlling parameters in the variation of s_u/u_*. Analytical models have been developed and supported by experiments for two limiting conditions: z/d < 0.02, -z/L < 0.5, and 0.02 << 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 wavenumber. The s_u/u_*  is shown to follow the much discussed logarithmic scaling s_u²/u_*² = B₁ - A₁ log (z/d), reconciled to Townsend's attached eddy hypothesis, 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 a k⁰, and k^-5/3 with a breakpoint predicted to occur 0.3 <kz <1. However, the mechanism of this shift of spectral behavior in between the domains of the two aforementioned formulations is still unknown and will be explored in future by the help of Large Eddy Simulations (LES), although signatures of this interesting behavior can be observed in experimental data. This useful theoretical framework is also extended to include stable stratification (Banerjee et al., 2015) but for fully developed turbulence (i.e. no laminarization or wave-like motion). However, this work 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.