Rayleigh-B\'enard convective motion of stratified fluids in the Earth's troposphere

TL;DR

This paper investigates Rayleigh-Bénard convective instability in the Earth's troposphere, considering both unbounded and bounded layers, and derives conditions and growth rates for instability through analytical and numerical methods.

Contribution

It provides a detailed analysis of convective instability in the troposphere, including new boundary conditions and quantification of vertical wave numbers for bounded layers.

Findings

Conditions for instability differ between unbounded and bounded layers.

Critical Rayleigh numbers and growth rates are derived.

Eigenvalue problem for bounded layers is well-posed.

Abstract

Recently, Kaladze and Misra [Phys. Scr. 99 (2024) 085013] showed that the tropospheric stratified fluid flows may be unstable by the effects of the negative temperature gradient and the temperature-dependent density inhomogeneity arising from the thermal expansion. They also predicted that the modification in the Brunt-V{\"a}is{\"a}l{\"a} frequency by the density inhomogeneity can lead to Rayleigh-B{\'e}nard convective instability in the tropospheric unbounded layers. The purpose of the present work is to revisit the Rayleigh-B{\'e}nard convective instability in more detail by considering both unbounded and bounded tropospheric layers. Starting from a set of fluid equations for incompressible neutral fluids with temperature gradients and using the Boussinesq approximation, we derive the general dispersion relations for Rayleigh-B{\'e}nard convective waves in unbounded and bounded tropospheric domains and analyze them with some particular cases both analytically and numerically. We show that the conditions for instability in these two cases significantly differ. Furthermore, we obtain and analyze the critical values of the Raleigh numbers and the expressions for the instability growth rates of thermal waves in the two cases. In the case of the bounded region, we also derive the necessary boundary conditions and note that the vertical wave number is quantified, and the corresponding eigenvalue problem is well-set.