Horizontal mean of the advected passive scalar for a simulation with a resolution of 180 × 90². The initial distribution of the scalar is shown as dashed line, the solid line represents the time-averaged profile for t ∈ [t(N_τconv = 9.9),t(N_τconv = 10.1)]. The blue shaded area corresponds to the minimal and maximal value of the passive scalar at the corresponding radius for the latest considered snapshot. The radii at which the absolute value of the radial derivative is largest are indicated by dots. They define the position of the top and bottom boundaries. The shaded orange area marks the convective region according to the stability criterion N² < 0. Vertical dashed lines denote the respective boundary widths which are defined in Sect. 5.6.