  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.
