A Theory for Nonprecipitating Convection between Two Parallel Plates. Part II: Nonlinear Theory and Cloud Field Organization

Abstract

In Part I, an idealized model of nonprecipitating moist convection in a shallow conditionally unstable layer of viscous and diffusive air between two parallel plates was introduced, and the ?linear? instability of an exactly saturated static state maintained by diffusion was investigated. If there are initially many clouds, the ?linear? theory predicted that weaker clouds are suppressed by the subsidence warming and drying from the ever-growing stronger clouds, and the average cloud spacing becomes arbitrarily large as time goes on. Each growing cloud is surrounded by compensating subsidence, which decreases away from the cloud with a characteristic decay scale Rs, the subsidence radius, which can be understood from gravity wave arguments. In Part II, fields of finite amplitude clouds are considered. An asymptotic analysis is performed in which the moist Rayleigh number Nc2 exceeds by only a small amount ? the value Nc02 necessary for the onset of convection. This leads to a nonlinear set of ?cloud field equations? which predict how the amplitudes and positions of all the clouds evolve in time. These equations predict a minimum stable cloud spacing ?c ≈ Rslog(??1). If the cloud spacing ? < ?c, slight differences in the strengths of neighboring clouds increase until the weaker clouds are suppressed. Unevenly spaced clouds drift until they become evenly spaced, ultimately resulting in a steady field of identical clouds with uniform spacing ? > ?c. Numerical experiments with dry stability Nd = Nc corroborate the conclusions from the cloud field equations when Nc2/Nc02 is less than ten. As Nc2 increases, the numerically determined ?c. becomes approximately 1.8Rs ≈ 1.8Nd. There is a second threshold spacing ?t ≈ 1.6Nd<?c not predicted by the asymptotic theory, below which a field of identical growing clouds is transient. This leads to two types of cloud field evolution. If Nc2/Nc02 is less than 10, all initial conditions lead to steady uniformly spaced fields of identical clouds. If Nc2/Nc02 is on the order of 10 or larger, a field of clouds initiated by horizontally homogeneous random buoyancy perturbations rapidly grows. While it is growing the subsidence radius around each cloud remains O(1); clouds are quite closely spaced. As the clouds mature, Rs increases rapidly to Nd. The clouds are spaced much closer than ?t apart, so they all dissipate. If the initial conditions are less random, however, so that a few widely spaced clouds break out first, these clouds inhibit the convection which later grows around them, ultimately become steady and drift toward a uniform spacing. In both cases there is no tendency toward cloud clustering. The steady cloud fields predicted by the model are probably never realized in the atmosphere due to other physical processes such as boundary layer forcing or precipitation, which favor small cloud spacings despite the large Rayleigh number. The primary conclusion that one can draw from the model is that compensating motions in the cloud layer are always competing with these other processes, tending to increase the spacing between convective clouds as subsidence-induced warming and drying suppresses the weaker circulations.