  We investigate the structures of the near-plate velocity and temperature
profiles at different horizontal positions along the conducting bottom (and
top) plate of a Rayleigh-B\'{e}nard convection cell, using two-dimensional (2D)
numerical data obtained at the Rayleigh number Ra=10^8 and the Prandtl number
Pr=4.4 of an Oberbeck-Boussinesq flow with constant material parameters. The
results show that most of the time, and for both velocity and temperature, the
instantaneous profiles scaled by the dynamical frame method [Q. Zhou and K.-Q.
Xia, Phys. Rev. Lett. 104, 104301 (2010) agree well with the classical
Prandtl-Blasius laminar boundary layer (BL) profiles. Therefore, when averaging
in the dynamical reference frames, which fluctuate with the respective
instantaneous kinematic and thermal BL thicknesses, the obtained mean velocity
and temperature profiles are also of Prandtl-Blasius type for nearly all
horizontal positions. We further show that in certain situations the
traditional definitions based on the time-averaged profiles can lead to
unphysical BL thicknesses, while the dynamical method also in such cases can
provide a well-defined BL thickness for both the kinematic and the thermal BLs.
