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 su 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 su2/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 su/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 su/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 su/u* is shown to follow the much discussed logarithmic scaling su2/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.