Balance of the Schwarzschild criterion in the vertical planes of the κ_const -disk (top) and the κ_BL-disk (remaining panels). The individual vertical planes are chosen to track the most prominent vertical flow at the given simulation time t (see the main text) and their azimuthal separations from the protoplanet location are given by panel labels (using multiples of the azimuthal span Δθ_H of the Hill sphere radius). The colour maps evaluate Eq. (15). Positive values indicate superadiabatic vertical temperaturegradients. The vertical velocity vector field is overlaid in the plots. The half-circles mark the overlap of a given plane with the Hill sphere (the planes in panels 3 and 5 do not overlap with the Hill sphere).