Fibonacci mean and golden section mean

AbstractIn this paper we show that there is a mapping D:M→DM on means such that if M is a Fibonacci mean so is DM, that if M is the harmonic mean, then DM is the arithmetic mean, and if M is a Fibonacci mean, then limn→∞DnM is the golden section mean

Power-Law Scaling in the Internal Variability of Cumulus Cloud Size Distributions due to Subsampling and Spatial Organization

In this study, the spatial structure of cumulus cloud populations is investigated using three-dimensional snapshots from large-domain LES experiments. The aim is to understand and quantify the internal variability in cloud size distributions due to subsampling effects and spatial organization. A set of idealized shallow cumulus cases is selected with varying degrees of spatial organization, including a slowly organizing marine precipitating case and five more quickly organizing diurnal cases over land. A subdomain analysis is applied, yielding cloud number distributions at sample sizes ranging from severely undersampled to nearly complete. A strong power-law scaling is found in the relation between cloud number variability and subdomain size, reflecting an inverse linear relation. Scaling subdomain size by cloud size yields a data collapse across time points and cases, highlighting the role played by cloud spacing in controlling the stochastic variability. Spatial organization acts on top of this baseline model by increasing the maximum cloud size and by enhancing the variability in the number of smallest clouds. This reflects that the smaller clouds start to live on top of larger-scale thermodynamic structures, such as cold pools, which favor or inhibit their formation. Compositing all continental cumulus cases suggests the existence of a prototype diurnal time dependence in the spatial organization. A simple stochastic expression for cloud number variability is proposed that is formulated in terms of two dimensionless groups, which allows objective estimation of the degree of spatial organization in simulated and observed cumulus cloud populations