Abstract
Convective clouds are a key driver of the earth's general circulation as well as producing important impacts on local weather. By investigating simplified systems that yield analytic solutions we hope to gain valuable insight into real world scenarios. We use the linearised Boussinesq form of the Navier-Stokes equations and the buoyancy equations derived for unsaturated and saturated air derived from the first law of thermodynamics as a basis to investigate the relationships between cloud spacing, cloud width and cloud growth rate for different choices of cloud scheme. We build on work by Bretherton who derived a cloud model which predicted realistic cloud solutions from a simplified model. We extend his work by first investigating the scenario of a fixed buoyancy frequency in the entire domain allowing cloud to form anywhere. By approaching the problem in this way we identify a critical relationship between growth rate and buoyancy frequency necessary for cloud solutions to form. We find this is an alternative form of Rayleigh-Benard convection with the same critical relationship posed in a different way. We then investigate the idea of buoyancy frequency being a smooth function across the cloud boundary. We solve this problem by expanding our chosen buoyancy frequency function as a Taylor series then use asymptotic methods to find a solution. By doing this we observe key differences in the predicted form of the clouds. In the case of periodic clouds with buoyancy frequency chosen to be a cosine function we find the relationship between growth rate and predicted cloud spacing has local growth rate maximums implying optimal cloud spacings for cloud growth.