The Deep Convection Approximation

Abstract

As has already been noted several times, atmospheric flows are low Mach number flows: M_∞ ² = U₀ ²/γRT_∞(0) ≪ 1. Because of this, if we wish to “avoid” the constraint (8.19) imposed in the Boussinesq approximation, it becomes necessary to analyze flows with very low Froude numbers since [see (2.63) and (2.65)]

$${\rm{Bo = }}\varepsilon \frac{{\gamma {\rm{M}}_\infty ^2}}{{{\rm{F}}{{\rm{r}}^{\rm{2}}}}} \Rightarrow \,{\rm{Fr}}\, \approx \,{{\rm{M}}_\infty } \ll 1\,,$$

(10.1)

if it is assumed that: Bo = gH/RT_∞(0), γ = c _p /c _v and ε = H/L are of the order unity.