Temperature profiles, plumes and spectra in the surface layer of convective boundary layers

We survey temperature patterns and heat transport in convective boundary layers (CBLs) from the perspective that these are emergent properties of far-from-equilibrium, complex dynamical systems. We introduce a two-temperature (2T) toy model to define the cross-sectional areas of plumes, and connect the scaling properties of temperature gradients, temperature variance and heat transport to this area. We examine temperature ($T$) probability density functions and $w$-$T$ joint probability density functions, $T$ spectra and $wT$ cospectra observed both within and above the surface friction layer. Here $w$ is vertical velocity. In our discussion of $T$ spectra and $wT$ cospectra we focus on the self-similarity property of the plumes and flux events above the SFL. We interpret the $z^{1/2}$ dependence of the mixed length scale for wavenumbers in the $T$ spectra as reflecting the cross-sectional areas of the plumes, and so with the $z^{-1/2}$ form of the temperature profile, where $z$ is observation height. We introduce new scaling results for $T$ spectra and $wT$ cospectra from within the surface friction layer (SFL), based on a data from the SLTEST experiment. We confirm earlier results showing that the scaling behaviours of $T$ spectra and $wT$ cospectra change for heights below $z/z_s<0.1$, where $z_s$ the height of the SFL, and come to display properties associated with random diffusion. We conclude by contrasting our interpretation of the role of buoyancy as a system-wide action in CBL flows with that of Richardson, whose ideas inform the current interpretation of the statistical fluid mechanics model of boundary-layer flows.