  In turbulent Rayleigh-B\'{e}nard (RB) convection, a transition to the
so-called ultimate regime, in which the boundary layers (BL) are of turbulent
type, has been postulated. Indeed, at very large Rayleigh number $Ra \approx
10^{13}-10^{14}$ a transition in the scaling of the global Nusselt number $Nu$
(the dimensionless heat transfer) and the Reynolds number with $Ra$ has been
observed in experiments and very recently in direct numerical simulations (DNS)
of two-dimensional (2D) RB. In this paper we analyse the local scaling
properties of the lateral temperature structure functions in the BLs of this
simulation of 2D RB, employing extended self-similarity (ESS) (i.e., plotting
the structure functions against each other, rather than only against the scale)
in the spirit of the attached eddy hypothesis, as we have recently introduced
for velocity structure functions in wall turbulence (Krug \textit{et al.},
\textit{J. Fluid Mech.}, vol. 830, 2017, pp. 797-819). We find no ESS scaling
below the transition and in the near wall region. However, beyond the
transition and for large enough wall distance $z^+ > 100$, we find clear ESS
behaviour, as expected for a scalar in a turbulent boundary layer. In striking
correspondence to the $Nu$ scaling, the ESS scaling region is negligible at $Ra
= 10^{11}$ and well developed at $Ra = 10^{14}$, thus providing strong evidence
that the observed transition in the global Nusselt number at $Ra \approx
10^{13}$ indeed is the transition from a laminar type BL to a turbulent type
BL. Our results further show that the relative slopes for scalar structure
functions in the ESS scaling regime are the same as for their velocity
counterparts extending their previously established universality. The findings
are confirmed by comparing to scalar structure functions in 3D turbulent
channel flow.
