  The logarithmic law of mean temperature profile has been observed in
different regions in Rayleigh-B\'enard turbulence. However, how thermal plumes
correlate to the log law of temperature and how the velocity profile changes
with pressure gradient are not fully understood. Here, we performed
three-dimensional simulations of Rayleigh-B\'enard turbulence in a slim-box
without the front and back walls with aspect ratio, $L:D:H=1:1/6:1$, in the
Rayleigh number $Ra=[1\times10^8, 1\times10^{10}]$ for Prandtl number $Pr=0.7$.
The velocity profile is successfully quantified by a two-layer function of a
stress length, $\ell_u^+\approx
\ell_0^+(z^+)^{3/2}\left[1+\left({z^+}/{z_{sub}^+}\right)^4\right]^{1/4}$, as
proposed by She et al. (She 2017), though neither a Prandtl-Blasius-Pohlhausen
type nor the log-law is seen in the viscous boundary layer. In contrast, the
temperature profile in the plume-ejecting region is logarithmic for all
simulated cases, being attributed to the emission of thermal plumes. The
coefficient of the temperature log-law, $A$ can be described by composition of
the thermal stress length $\ell^*_{\theta}$ and the thicknesses of thermal
boundary layer $z^*_{sub}$ and $z^*_{buf}$, i.e. $A\simeq
z^*_{sub}/\left(\ell^*_{\theta 0}{z^*_{buf}}^{3/2}\right)$. The adverse
pressure gradient responsible for turning the wind direction contributes to
thermal plumes gathering at the ejecting region and thus the log-law of
temperature profile. The Nusselt number scaling and local heat flux of the
present simulations are consistent with previous results in confined cells.
Therefore, the slim-box RBC is a preferable system for investigating in-box
kinetic and thermal structures of turbulent convection with the large-scale
circulation on a fixed plane.
