Convection in a Conditionally Unstable Atmosphere: A Re-investigation of BJERKNES' Slice Method

Abstract

A general formulation of the buoyancy budget in a horizontal slice, in the middle of a conditionally unstable atmosphere, permits a comparison of various convection patterns, and provides a natural extension of BJERKNES' (1938) theory. Here we consider centre-up, centre-down, and roll patterns. Lateral entrainment between up- and downdraught region is assumed to consist of two parts: background mixing and mixing induced by the convective motions themselves. Due to the latter nonlinearity, finite-amplitude convection is possible. Linear stability analysis of the state of rest shows that the centre-up pattern is the most unstable configuration. Non-zero steady states are also compared. It turns out that the centre-down pattern has the smallest amplitude, while the centre-up pattern 'performs best'. The results suggest that a centre-down pattern can only persist when background mixing is small. This agrees with the fact that open cells, qualitatively similar to the centre-down pattern analysed here, are normally found over homogeneous water surfaces when wind shear is not too large. Several authors have suggested that maximizing the upward heat flux can be used to determine a preferred horizontal length scale for convection. The results presented here cast some doubt on the applicability of such an extremal principle, because open cells, so frequently observed in reality, have the smallest upward heat flux.