A study of convective instability in the shallow troposphere: a non-Boussinesq approach

Abstract:

In this work, we present a simulation study of the convective instability (CI) which is excited in the presence of a negative temperature gradient in the dry troposphere. Using incompressible-inviscid hydrodynamic equations and laws of thermodynamics, we derive a growth equation for the wind w_y, explicitly containing a growth term with frequency equals to the modified Brunt-Vaisala frequency (ω_b) and a non-local term. A general expression for the modified Brunt-Vaisala frequency is derived for the dry-shallow troposphere, retaining the diabatic compression and non-linearity of the density (Diabatic-non-Boussinesq, DNB). This expression is also analyzed under adiabatic compressibility for two approaches: without the density linearizatin (Adiabatic-non-Boussinesq, ANB) and with the density linearization (Boussinesq, OB). The DNB approach is desirable in the dry troposphere where the environmental lapse rate (γ_e) is weaker than the adiabatic lapse rate (γ_ad). Under the DNB approach, ω_b is found to be modified from its OB counterpart such that stabilizing γ_ad is replaced by a modified lapse rate (η-1) γ_ac. Here, (γ_ac,η) are the auto-convective lapse rate and the ratio of hydrostatic density to the total density. It is shown that under ANB approach, η becomes equal to the ratio of specific heats (γ). We show by simulation of the CI that as η increases from unity towards γ within a heated fluid parcel, the CI changes from DNB to ANB type and γ_e approaches toward γ_ad. Notably, the CI growth remains positive during DNB phase for the initial temperature profile having γ_e < γ_ad. The present study, thus, proposes an alternative mechanism for the positive growth of the CI under an initial adiabatically stable dry troposphere.