  We theoretically investigate the spatial dependence of the order parameter of
the two-dimensional semi-infinite Ising model with a free surface at or above
the bulk critical temperature. Special attention is paid to the influence of a
surface magnetic field $h_1$ and the crossover between the fixed points at
h_1=0 and h_1=infinity. The sharp increase of the magnetization m(z) close to
the boundary generated by a small h_1, which was found previously by the
present authors in the three-dimensional model, is also seen in two dimensions.
There, however, the universal short-distance power law is modified by a
logarithm. By means of a phenomenological scaling analysis, the short-distance
behavior can be related to the logarithmic dependence of the surface
magnetization on h_1. Our results, which are corroborated by Monte Carlo
simulations, provide a deeper understanding of the existing exact results
concerning the local magnetization and relate the short-distance phenomena in
two dimensions to those in higher dimensionality.
