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In this work, we study the growth characteristics of the convectiveinstability (CI) in the dry troposphere by relaxing the adiabaticcompressibility condition of Oberbeck-Boussinesq (OB) approach. We derive a newnon-adiabatic-Boussinesq (NAB) expression for the modified Brunt-Vaisalafrequency $(\omega_b)$, without considering the adiabatic compressibilitycondition of OB approach. This NAB expression reduces to the knownOberbeck-Boussinesq (OB) expression under adiabatic compressibility condition.The NAB expression of $\omega_b$ is found to be modified from its OBcounterpart such that the stabilizing adiabatic lapse rate in OB expression isreplaced by a modified non-adiabatic lapse rate given as $\left(\eta - 1\right)$ times the auto-convective lapse rate. Here $\eta$ is the ratio ofhydrostatic density to the total density. We perform numerical experiments ofCI for the conditionally stable troposphere i.e for the troposphere that hasthe environmental lapse rate negative but smaller than the adiabatic lapserate. A novel feature of the present study is that the CI grows under proposedNAB approach in spite the conditionally stable condition and remains suppressedunder OB approach. The present study, thus, proposes an alternative NABapproach for the positive growth of the CI in the dry troposphere for which theCI is conditionally stable under OB approach.

The growth of dry convection in the conditionally stable troposphere: Non-adiabatic effects