Logarithm of the density (left) and gas pressure (right) as a function of the normalized radial distance to the center of the sunspot ξ = r/R_spot at four different vertical heights z = 0, 512, 1024, 1528 km. Larger z values correspond to higher atmospheric layers. Black circles correspond to the azimuthal- or ϕ-averages of the MHD simulations (Sect. 2). Red curves are obtained through fourth-order polynomial approximations to the black circles.