The growth of dry convection in the conditionally stable troposphere:   Non-adiabatic effects
  In this work, we study the growth characteristics of the convective instability (CI) in the dry troposphere by relaxing the adiabatic compressibility condition of Oberbeck-Boussinesq (OB) approach. We derive a new non-adiabatic-Boussinesq (NAB) expression for the modified Brunt-Vaisala frequency $(\omega_b)$, without considering the adiabatic compressibility condition of OB approach. This NAB expression reduces to the known Oberbeck-Boussinesq (OB) expression under adiabatic compressibility condition. The NAB expression of $\omega_b$ is found to be modified from its OB counterpart such that the stabilizing adiabatic lapse rate in OB expression is replaced by a modified non-adiabatic lapse rate given as $\left(\eta - 1 \right)$ times the auto-convective lapse rate. Here $\eta$ is the ratio of hydrostatic density to the total density. We perform numerical experiments of CI for the conditionally stable troposphere i.e for the troposphere that has the environmental lapse rate negative but smaller than the adiabatic lapse rate. A novel feature of the present study is that the CI grows under proposed NAB approach in spite the conditionally stable condition and remains suppressed under OB approach. The present study, thus, proposes an alternative NAB approach for the positive growth of the CI in the dry troposphere for which the CI is conditionally stable under OB approach. 
E.A. Kherani, B.R. Tiwari and J.H.A. Sobral
physics.ao-ph